Prove term relationship

A question is this type if and only if it asks you to prove or show that a specific relationship holds between terms of one or more geometric progressions (e.g., show that one nth term equals a multiple of another).

4 questions · Standard +0.1

1.04i Geometric sequences: nth term and finite series sum
Sort by: Default | Easiest first | Hardest first
OCR C2 2005 January Q9
11 marks Standard +0.3
9 A geometric progression has first term \(a\), where \(a \neq 0\), and common ratio \(r\), where \(r \neq 1\). The difference between the fourth term and the first term is equal to four times the difference between the third term and the second term.
  1. Show that \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1 = 0\).
  2. Show that \(r - 1\) is a factor of \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\). Hence factorise \(r ^ { 3 } - 4 r ^ { 2 } + 4 r - 1\).
  3. Hence find the two possible values for the ratio of the geometric progression. Give your answers in an exact form.
  4. For the value of \(r\) for which the progression is convergent, prove that the sum to infinity is \(\frac { 1 } { 2 } a ( 1 + \sqrt { } 5 )\).
OCR MEI Paper 3 2018 June Q13
3 marks Moderate -0.5
13 Consider a geometric sequence in which all the terms are positive real numbers. Show that, for any three consecutive terms of this sequence, the middle one is the geometric mean of the other two.
CAIE P1 2023 June Q9
8 marks Standard +0.3
The second term of a geometric progression is 16 and the sum to infinity is 100.
  1. Find the two possible values of the first term. [4]
  2. Show that the \(n\)th term of one of the two possible geometric progressions is equal to \(4^{n-2}\) multiplied by the \(n\)th term of the other geometric progression. [4]
OCR MEI C2 Q7
10 marks Standard +0.3
A geometric progression has first term \(a\) and common ratio \(r\). The second term is 6 and the sum to infinity is 25.
  1. Write down two equations in \(a\) and \(r\). Show that one possible value of \(a\) is 10 and find the other possible value of \(a\). Write down the corresponding values of \(r\). [7]
  2. Show that the ratio of the \(n\)th terms of the two geometric progressions found in part (i) can be written as \(2^{n-2} : 3^{n-2}\). [3]