Geometric Sequence Problem

00:00 6 marks

A geometric sequence has first term \(u₁ = 120\) and common ratio \(r = 0.9\).

(a) Find the value of the fifth term, \(u₅\). [2]

(b) Find the sum of the first 10 terms of the sequence. [2]

(c) Find the smallest number of terms, \(n\), for which the sum of the first \(n\) terms, \(Sₙ\), is greater than 1000. [2]

3 minutes 2 marks

Find the value of the fifth term, \(u₅\). [2]

Hint Hint

Use the formula \(uₙ = u₁ \times r^{(n-1)}\) and substitute n = 5.

78.7

3 minutes 2 marks

Find the sum of the first 10 terms of the sequence. [2]

Hint Hint

Use \(Sₙ = \frac{120(1 – r^n)}{1 – r}\) with n = 10.

782

4 minutes 2 marks

Find the smallest number of terms, \(n\), for which the sum \(Sₙ\) is greater than 1000. [2]

Hint Hint

Set up the inequality \(\frac{120(1 – 0.9^n)}{0.1} > 1000\) and solve for n using logarithms.