The common ratio of a geometric progression is \(r\). The first term of the progression is \((r^2 - 3r + 2)\) and the sum to infinity is \(S\).
A geometric progression has a first term of 6 and a sum to infinity of 18. A new geometric progression is formed by squaring each of the terms of the original progression. Find the sum to infinity of the new progression.
Each year a school allocates a sum of money for the library. The amount allocated each year increases by 2.5% of the amount allocated the previous year. In 2005 the school allocated $2000. Find the total amount allocated in the years 2005 to 2014 inclusive.
Two convergent geometric progressions, P and Q, have the same sum to infinity. The first and second terms of P are 6 and 6r respectively. The first and second terms of Q are 12 and -12r respectively. Find the value of the common sum to infinity.
A geometric progression is such that the third term is 8 times the sixth term, and the sum of the first six terms is 31\(\frac{1}{2}\). Find
The sum of the 1st and 2nd terms of a geometric progression is 50 and the sum of the 2nd and 3rd terms is 30. Find the sum to infinity.
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.
Assume instead that 10 litres of water are lost on the first day and that the amount of water lost increases by 10% on each succeeding day. Find what percentage of the original 2000 litres is left in the tank at the end of the 30th day after filling.
The first term of a geometric progression in which all the terms are positive is 50. The third term is 32. Find the sum to infinity of the progression.
The first term of a progression is \(4x\) and the second term is \(x^2\).
For the case where the progression is geometric with a sum to infinity of 8, find the third term.
The first, second and third terms of a geometric progression are \(2k + 6\), \(2k\) and \(k + 2\) respectively, where \(k\) is a positive constant.
(i) Find the value of \(k\).
(ii) Find the sum to infinity of the progression.
The second term of a geometric progression is 16 and the sum to infinity is 100.
(a) Find the two possible values of the first term.
(b) Show that the nth term of one of the two possible geometric progressions is equal to \(4^{n-2}\) multiplied by the nth term of the other geometric progression.
The third and fourth terms of a geometric progression are \(\frac{1}{3}\) and \(\frac{2}{9}\) respectively. Find the sum to infinity of the progression.
A geometric progression in which all the terms are positive has sum to infinity 20. The sum of the first two terms is 12.8. Find the first term of the progression.
A geometric progression has first term \(a\) \((a \neq 0)\), common ratio \(r\) and sum to infinity \(S\). A second geometric progression has first term \(a\), common ratio \(2r\) and sum to infinity \(3S\). Find the value of \(r\).
In a geometric progression, the sum to infinity is equal to eight times the first term. Find the common ratio.
The second and third terms of a geometric progression are 48 and 32 respectively. Find the sum to infinity of the progression.
A geometric progression has first term a, common ratio r and sum to infinity 6. A second geometric progression has first term 2a, common ratio r2 and sum to infinity 7. Find the values of a and r.
The third term of a geometric progression is four times the first term. The sum of the first six terms is k times the first term. Find the possible values of k.
The third term of a geometric progression is -108 and the sixth term is 32. Find
The first term of a geometric progression is \(5\frac{1}{3}\) and the fourth term is \(2\frac{1}{4}\). Find
(i) the common ratio,
(ii) the sum to infinity.