The functions \(f\) and \(g\) are defined by
\(f(x)=\ln(3x+2),\quad x\gt-\frac23,\)
and
\(g(x)=e^{2x}-4,\quad x\in\mathbb{R}.\)
(i) Solve \(gf(x)=5\).
(ii) Find \(f^{-1}(x)\).
(iii) Solve \(f^{-1}(x)=g(x)\).
A progression has first term a and second term \(\frac{a^2}{a+2}\), where a is a positive constant.
For the case where the progression is arithmetic and \(a = 6\), determine the least value of n required for the sum of the first n terms to be less than -480.
The sum, \(S_n\), of the first \(n\) terms of an arithmetic progression is given by
\(S_n = n^2 + 4n\).
The \(k\)th term in the progression is greater than 200.
Find the smallest possible value of \(k\).
The nth term of an arithmetic progression is \(\frac{1}{2}(3n - 15)\).
Find the value of n for which the sum of the first n terms is 84.
The sum of the first nine terms of an arithmetic progression is 117. The sum of the next four terms is 91.
Find the first term and the common difference of the progression.
Over a 21-day period an athlete prepares for a marathon by increasing the distance she runs each day by 1.2 km. On the first day she runs 13 km.
(i) Find the distance she runs on the last day of the 21-day period.
(ii) Find the total distance she runs in the 21-day period.
In an arithmetic progression, the sum of the first ten terms is equal to the sum of the next five terms. The first term is \(a\).
(i) Show that the common difference of the progression is \(\frac{1}{3}a\).
(ii) Given that the tenth term is 36 more than the fourth term, find the value of \(a\).
In another case, p and 2p are the first and second terms respectively of an arithmetic progression. The nth term is 336 and the sum of the first n terms is 7224. Write down two equations in n and p and hence find the values of n and p.
In an arithmetic progression the first term is a and the common difference is 3. The nth term is 94 and the sum of the first n terms is 1420. Find n and a.
The first term of a series is 6 and the second term is 2.
For the case where the series is an arithmetic progression, find the sum of the first 80 terms.
The nth term of a progression is p + qn, where p and q are constants, and Sn is the sum of the first n terms.
An arithmetic progression has first term \(-12\) and common difference \(6\). The sum of the first \(n\) terms exceeds \(3000\). Calculate the least possible value of \(n\).
The circumference round the trunk of a large tree is measured and found to be 5.00 m. After one year the circumference is measured again and found to be 5.02 m.
Given that the circumferences at yearly intervals form an arithmetic progression, find the circumference 20 years after the first measurement.
The sum of the first n terms of an arithmetic progression is \(\frac{1}{2}n(3n + 7)\). Find the 1st term and the common difference of the progression.
The first two terms of an arithmetic progression are 15 and 19 respectively. The first two terms of a second arithmetic progression are 420 and 415 respectively. The two progressions have the same sum of the first n terms. Find the value of n.
The first two terms of an arithmetic progression are 16 and 24. Find the least number of terms of the progression which must be taken for their sum to exceed 20,000.
An arithmetic progression has a first term of 32, a 5th term of 22 and a last term of -28. Find the sum of all the terms in the progression.
A cyclist completes a long-distance charity event across Africa. The total distance is 3050 km. He starts the event on May 1st and cycles 200 km on that day. On each subsequent day he reduces the distance cycled by 5 km.
(i) How far will he travel on May 15th?
(ii) On what date will he finish the event?
A water tank holds 2000 litres when full. A small hole in the base is gradually getting bigger so that each day a greater amount of water is lost.
On the first day after filling, 10 litres of water are lost and this increases by 2 litres each day.
(a) How many litres will be lost on the 30th day after filling?
(b) The tank becomes empty during the nth day after filling. Find the value of n.
The 12th term of an arithmetic progression is 17 and the sum of the first 31 terms is 1023. Find the 31st term.