An arithmetic progression has first term \(t\) and common difference \(1.5\). The 4th, 8th and 20th terms of this arithmetic progression form the 1st, 2nd and 3rd terms of a geometric progression.

(a) Find the value of \(t\).

(b) Find the common ratio of the geometric progression.

(a) An arithmetic progression has first term \(a\) and common difference \(d\). Given that \(S_{20}=3S_{10}\), find \(a\) in terms of \(d\).

(b) A geometric progression, A, has common ratio \(r\), where \(|r|\lt 1\). The terms of this progression are \(a_1,a_2,a_3,\ldots\).

Another geometric progression, B, has terms \(b_1,b_2,b_3,\ldots\), where \(b_1=a_2\), \(b_2=a_4\), \(b_3=a_6,\ldots\).

The sum to infinity of A is \(S_A\), and the sum to infinity of B is \(S_B\). Find \(\frac{S_B}{S_A}\) in terms of \(r\). Give your answer in its simplest form.

(a) The first term of an arithmetic progression is \(3\). The sum of the first 10 terms is four times the sum of the first 5 terms. Find the common difference.

(b) The first, second and fifth terms of another arithmetic progression are the first, second and third terms of a geometric progression. The first term is non-zero. Find the common ratio of the geometric progression, where the common ratio is not \(1\).

The first three terms of an arithmetic progression can be written as \(2 \ln \left(x^{3}\right), \quad 5 \ln \left(x^{2}\right), \quad 2 \ln \left(x^{7}\right) .\) (a) Given that \(x\gt 1\), find the least number of terms for the sum of this progression to be greater than \(43 \ln \left(x^{24}\right)\) (b) Given that the 25th term of this progression is equal to 408 , find the exact value of \(x\).

(a) The 1st term of an arithmetic progression is 9 . The last term of this progression is 159 . The sum of all the terms is 2604 .

The 12th term of this arithmetic progression is the 1st term of a geometric progression. The 8th term of this arithmetic progression is the 2nd term of the geometric progression.

Find the sum of the first 6 terms of the geometric progression.

(b) A different geometric progression has 1st term \(\sin \theta\).

The common ratio of this progression is \(\cos \theta\) where \(45^{\circ} \leqslant \theta \leqslant 135^{\circ}\). (i) Show that this progression has a sum to infinity.

(ii) Show that this sum to infinity can be written as \(\operatorname{cosec} \theta+\cot \theta\).

(a) The first 3 terms of an arithmetic progression are \(2 \tan 2 x, 5 \tan 2 x, 8 \tan 2 x\). Find the values of \(x\), where \(-180^{\circ} \leqslant x \leqslant 180^{\circ}\), for which the sum to 30 terms is \(455 \sqrt{3}\).

(b) The first 3 terms of a geometric progression are \(5 \cos ^{2}\left(\theta-\frac{\pi}{2}\right), \quad 20 \cos ^{4}\left(\theta-\frac{\pi}{2}\right), \quad 80 \cos ^{6}\left(\theta-\frac{\pi}{2}\right), \quad \text { where }-\frac{\pi}{6} \leqslant \theta \leqslant \frac{7 \pi}{6} .\)

Find the values of \(\theta\) for which this geometric progression has a sum to infinity.

(a) The first 3 terms of an arithmetic progression are \(\log _{x} 3, \log _{x} 81, \log _{x} 2187\). Find the sum to \(n\) terms, giving your answer in the form \(k \log _{x} 3\), where \(k\) is in terms of \(n\).

(b) The first 3 terms of a geometric progression are \(1,3 \tan ^{2} \theta, 9 \tan ^{4} \theta\), for \(0\lt \theta\lt \frac{\pi}{2}\). Find the values of \(\theta\) for which this geometric progression has a sum to infinity.

(a) The first three terms of an arithmetic progression are \(\lg \theta^{2}, \lg \theta^{5}\) and \(\lg \theta^{8}\). (i) Given that the sum to \(n\) terms of this progression is \(4732 \lg \theta\), find the value of \(n\).

(ii) This sum is equal to -14196 . Find the exact value of \(\theta\).

(b) The first three terms of a geometric progression are \(\lg \phi^{3}, \lg \phi\) and \(\lg \phi^{\frac{1}{3}}\). (i) Determine whether this geometric progression has a sum to infinity.

(ii) Find the \(n\)th term of this geometric progression, giving your answer in the form \(3^{A} \lg \phi\), where \(A\) is a function of \(n\). (iii) Find the value of \(\phi\), given that the 20th term is \(3^{-18}\).

(a) The first 3 terms of an arithmetic progression are \(3 \sin 2 x, 5 \sin 2 x, 7 \sin 2 x\). (i) Show that the sum to \(n\) terms of this arithmetic progression can be written in the form \(n(n+a) \sin 2 x\), where \(a\) is a constant. (ii) Given that \(x=\frac{2 \pi}{3}\), find the exact sum of the first 20 terms.

(b) The first 3 terms of a geometric progression are \(\ln 2 y, \quad \ln 4 y^{2}, \quad \ln 16 y^{4}\). (i) Find the \(n\)th term of this geometric progression.

(ii) Find the sum to \(n\) terms of this geometric progression, giving your answer in its simplest form.

(c) The first 3 terms of a different geometric progression are \(\left(2 w-\frac{1}{4}\right),\left(2 w-\frac{1}{4}\right)^{2},\left(2 w-\frac{1}{4}\right)^{3}\). Find the values of \(w\) for which this geometric progression has a sum to infinity.

(a) In an arithmetic progression, the first term is \(a\) and the common difference is \(d\). The sum of the first three terms of this arithmetic progression is 42 . The product of the first three terms of this arithmetic progression is -6720 . (i) Show that \(a(a+2 d)=-480\).

(ii) Hence, given that \(a\) is positive, find the values of \(a\) and \(d\).

(b) In a geometric progression, the 3 rd term is \(\frac{\mathrm{e}^{4 x}}{4}\) and the 10 th term is \(\frac{\mathrm{e}^{11 x}}{512}\). Find the first term and the common ratio.

(a) In an arithmetic progression, the sum of the first 30 terms is -1065 . The sum of the next 20 terms is -2210 . Find the first term and the common difference.

(b) A geometric progression is such that the first term is 4 and the sum of the first three terms is 7 . Find the two possible values of the common ratio and find the sum to infinity for the convergent progression.

(a) In an arithmetic progression the 5th term is \(11\). The 7th term is three times the 2nd term. Find the 1st term and the common difference.

(b) An arithmetic progression and a geometric progression both have first term \(3\). The 2nd term of the arithmetic progression is equal to the 3rd term of the geometric progression. The 6th term of the arithmetic progression is equal to the 5th term of the geometric progression. Given that the common ratio of the geometric progression is greater than \(1\), find the common difference of the arithmetic progression and the common ratio of the geometric progression.

(a) An arithmetic progression has twelve terms. The sum of the first three terms is \(-36\) and the sum of the last three terms is \(72\). Find the first term and the common difference.

(b) The first three terms of a geometric progression are \(1\), \(1.2\) and \(1.44\). Find the smallest value of \(n\) such that the sum of the first \(n\) terms is greater than \(500\).

(a) The terms \(\ln q\), \(\ln q^4\), \(\ln q^7\), where \(q\) is positive, are the first three terms of an arithmetic progression. The sum of the first \(n\) terms of this progression is \(4845\ln q\). Find the value of \(n\).

(b) The terms \(p^{3x}\), \(p^x\), \(p^{-x}\), where \(p\) is positive, are the first three terms of a geometric progression. Find the \(n\)th term of this progression in the form \(p^{(a+bn)x}\), where \(a\) and \(b\) are integers.

(c) The first three terms of a geometric progression are

\(\displaystyle \frac43\cos^2 3\theta,\quad \frac{16}{9}\cos^4 3\theta,\quad \frac{64}{27}\cos^6 3\theta,\)

where \(0\lt\theta\lt\frac{\pi}{3}\). Find the set of values of \(\theta\) for which this progression has a sum to infinity.

(a) The first three terms of an arithmetic progression are \((2x+1)\), \(4(2x+1)\) and \(7(2x+1)\), where \(x\ne-\frac12\).

(i) Show that the sum to \(n\) terms can be written in the form \(\frac n2(2x+1)(An+B)\), where \(A\) and \(B\) are integers to be found.

(ii) Given that the sum to \(n\) terms is \((54n+37)(2x+1)\), find the value of \(n\).

(iii) Given also that the sum to \(n\) terms in part (ii) is equal to \(1017.5\), find the value of \(x\).

(b) The first three terms of a geometric progression are \((2y+1)\), \(3(2y+1)^2\) and \(9(2y+1)^3\), where \(y\ne-\frac12\). Given that the \(n\)th term of the progression is equal to 4 times the \((n+2)\)th term, find the possible values of \(y\), giving your answers as fractions.

(c) The first three terms of a different geometric progression are \(\sin\theta\), \(2\sin^3\theta\) and \(4\sin^5\theta\), for \(0\lt\theta\lt\frac{\pi}{2}\). Find the values of \(\theta\) for which the progression has a sum to infinity.

An arithmetic progression \(A\) has first term \(a\) and common difference \(d\). The second, fourteenth and seventeenth terms of \(A\) form the first three terms of a convergent geometric progression \(G\) with common ratio \(r\).

(a)(i) Given that \(d\ne0\), find two expressions for \(r\) in terms of \(a\) and \(d\), and hence show that \(a=-17d\).

(a)(ii) Find \(r\).

(b) The first term of \(G\) is \(q\), and the sum to infinity of \(G\) is \(\frac{256}{3}\). Find the sum of the first 20 terms of \(A\).

(a) The first three terms of an arithmetic progression are

\(-3\tan\frac{\theta}{2},\quad -\tan\frac{\theta}{2},\quad \tan\frac{\theta}{2},\)

where \(0\lt \theta\lt \frac12\pi\).

Given that the 12th term is \(\frac{19\sqrt3}{3}\), find

(i) the value of \(\theta\),

(ii) the sum of the first 10 terms.

(b) The first three terms of a geometric progression are

\(\frac{1}{16}\operatorname{cosec}^4\phi,\quad \frac14\operatorname{cosec}^2\phi,\quad 1,\)

where \(-\frac12\pi\lt \phi\lt \frac12\pi\).

(i) Given that the sum of the 3rd and 4th terms is 4, find the possible values of \(\phi\).

(ii) Determine whether this geometric progression has a sum to infinity.

(a) The first three terms of an arithmetic progression are \(\sin3x\), \(5\sin3x\), \(9\sin3x\). Find the exact values of \(x\), where \(0\le x\le\frac{\pi}{2}\), for which the sum to twenty terms is equal to 390.

(b) The first three terms of a geometric progression are \(20\cos y\), \(10\cos^2y\), \(5\cos^3y\).

(i) Explain why this progression has a sum to infinity.

(ii) Find the value of \(y\), where \(y\) is in radians and \(0\lt y\lt 2\), for which the sum to infinity is 9. Give your answer correct to 2 decimal places.

(a) A geometric progression has first term \(64\) and common ratio \(0.5\).

(i) Find the 10th term.

(ii) Find the sum of the first 10 terms.

(iii) Find the sum to infinity.

(b) An arithmetic progression has first term \(a\), common difference \(d\), and sum of the first \(n\) terms \(S_n\). It is given that

\(S_{20}-400=2S_{10}\)

and

\(u_1:u_6=1:5.\)

Find the sum of the first 3 terms of this arithmetic progression.

(a) The first three terms of an arithmetic progression are \(\lg3\), \(3\lg3\), \(5\lg3\). Given that the sum to \(n\) terms is \(256\lg81\), find \(n\).

(b) The first three terms of a geometric progression are \(\ln256\), \(\ln16\), \(\ln4\). Find the sum to infinity in the form \(p\ln2\).

(a) A geometric progression has first term \(a\) and common ratio \(r\), where \(r\gt 0\). The second term of this progression is 8. The sum of the third and fourth terms is 160.

(i) Show that \(r\) satisfies the equation \(r^2+r-20=0\).

(ii) Find the value of \(a\).

(b) An arithmetic progression has first term \(p\) and common difference 2. The \(q\)th term of this progression is 14. A different arithmetic progression has first term \(p\) and common difference 4. The sum of the first \(q\) terms of this progression is 168. Find the values of \(p\) and \(q\).

(a) An arithmetic progression has first term \(a\) and common difference \(d\). The sum of the first 20 terms is 1100 and the sum of the first 70 terms is 14350. Find the 12th term.

(b) The first three terms of a geometric progression are \(x+6\), \(x-9\) and \(\frac12(x+1)\).

Show that \(x^2-43x+156=0\), and hence show that the sum to infinity exists for each possible value of \(x\).

(a) A geometric progression has first term \(10\) and sum to infinity \(6\).

(i) Find the common ratio of this progression.

(ii) Hence find the sum of the first 7 terms, giving your answer correct to 2 decimal places.

(b) The first three terms of an arithmetic progression are \(\log_x3\), \(\log_x(3^2)\), \(\log_x(3^3)\).

(i) Find the common difference of this progression.

(ii) Find, in terms of \(n\) and \(\log_x3\), the sum to \(n\) terms of this progression. Simplify your answer.

(iii) Given that the sum to \(n\) terms is \(3081\log_x3\), find the value of \(n\).

(iv) Hence, given that the sum to \(n\) terms is also equal to \(1027\), find the value of \(x\).

(a) The first three terms of an arithmetic progression are \(-4\), \(8\) and \(20\).

Find the smallest number of terms of this progression which have a sum greater than \(2000\).

(b) The \(7\)th term of a geometric progression is \(27\), and the \(9\)th term is \(243\). The common ratio is positive.

(i) Find the first term and the common ratio.

(ii) Find the \(30\)th term, giving your answer as a power of \(3\).

(c) Explain why the geometric progression

\(1,\ \sin\theta,\ \sin^2\theta,\ \sin^3\theta,\ldots\)

has a sum to infinity for \(-\frac{\pi}{2}\lt \theta\lt \frac{\pi}{2}\).

The \(2\)nd, \(8\)th and \(44\)th terms of an arithmetic progression form the first three terms of a geometric progression. In the arithmetic progression, the first term is \(1\) and the common difference is positive.

(a)

(i) Show that the common difference of the arithmetic progression is \(5\).

(ii) Find the sum of the first \(20\) terms of the arithmetic progression.

(b)

(i) Find the \(5\)th term of the geometric progression.

(ii) Explain whether or not the sum to infinity of this geometric progression exists.

(a) The first three terms of an arithmetic progression are

\(\frac1p,\quad \frac1q,\quad -\frac1q.\)

(i) Show that the common difference can be written as

\(-\frac2{3p}.\)

(ii) The \(10\)th term of the progression is \(\frac{k}{p}\), where \(k\) is a constant. Find \(k\).

(b) The sum to infinity of a geometric progression is \(8\). The second term of the progression is \(\frac32\). Find the two possible values of the common ratio.

(a) Jess runs on 5 days each week to prepare for a race. In week 1, every run is \(2\text{ km}\). In week 2, every run is \(2.5\text{ km}\). In week 3, every run is \(3\text{ km}\). Jess increases the distance of the run by \(0.5\text{ km}\) every week.

(i) Find the week in which Jess runs \(16\text{ km}\) on each of the 5 days.

(ii) Find the total distance Jess will have run by the end of week 8.

(b) Kyle also runs on 5 days each week to prepare for a race. In week 1, every run is \(2\text{ km}\). In week 2, every run is \(2.5\text{ km}\). In week 3, every run is \(3.125\text{ km}\). The distances he runs each week form a geometric progression.

(i) Find the common ratio of the geometric progression.

(ii) Find the first week in which Kyle will run more than \(16\text{ km}\) on each of the 5 days.

(iii) Find the total distance Kyle will have run by the end of week 8.

(a) The first three terms of an arithmetic progression are \(x\), \(5x-4\) and \(8x+2\). Find \(x\) and the common difference.

(b) The first three terms of a geometric progression are \(y\), \(5y-4\) and \(8y+2\).

(i) Find the two possible values of \(y\).

(ii) For each of these values of \(y\), find the corresponding value of the common ratio.

(a) The sum of the first two terms of a geometric progression is \(10\). The third term is \(9\). Find the possible values of the common ratio and the corresponding first terms. For the convergent progression, find the sum to infinity.

(b) An arithmetic progression has first term \(-10\) and fourth term \(14\). Find the value of

\(u_{100}+u_{101}+\cdots+u_{200}.\)

(a) An arithmetic progression has a second term of \(-14\) and a sum to \(21\) terms of \(84\). Find the first term and the \(21\)st term of this progression.

(b) A geometric progression has a second term of \(27p^2\) and a fifth term of \(p^5\). The common ratio, \(r\), is such that \(0\lt r\lt1\).

(i) Find \(r\) in terms of \(p\).

(ii) Hence find, in terms of \(p\), the sum to infinity of the progression.

(iii) Given that the sum to infinity is \(81\), find the value of \(p\).

(a) An arithmetic progression has a first term of \(7\) and a common difference of \(0.4\). Find the least number of terms so that the sum of the progression is greater than \(300\).

(b) The sum of the first two terms of a geometric progression is \(9\) and its sum to infinity is \(36\). Given that the terms of the progression are positive, find the common ratio.

(a) The first 5 terms of a sequence are

\(4,\quad -2,\quad 1,\quad -0.5,\quad 0.25.\)

(i) Find the 20th term of the sequence.

(ii) Explain why the sum to infinity exists for this sequence and find the value of this sum.

(b) The tenth term of an arithmetic progression is 15 times the second term. The sum of the first 6 terms of the progression is 87.

(i) Find the common difference of the progression.

(ii) For this progression, the \(n\)th term is 6990. Find the value of \(n\).

(a) An arithmetic progression has a second term of \(8\) and a fourth term of \(18\). Find the least number of terms for which the sum of this progression is greater than \(1560\).

(b) A geometric progression has a sum to infinity of \(72\). The sum of the first \(3\) terms of this progression is \(\dfrac{333}{8}\).

(i) Find the value of the common ratio.

(ii) Hence find the value of the first term.

The sum of the first 4 terms of an arithmetic progression is \(38\). The sum of the next 4 terms is \(86\). Find the first term and common difference of the arithmetic progression.

The third term of a geometric progression is \(12\), and the sixth term is \(-96\). Find the sum of the first 10 terms of the geometric progression.