Sum of infinite convergent geometric sequences practice questions.

An infinite geometric series has first term 12 and common ratio −1/3. Calculate its sum to infinity.

• 18
• 8
• 12
• 9

Answer: 9

S∞ = u₁/(1 − r) = 12/(1 − (−1/3)) = 12/(4/3) = 9. Watch the sign: subtracting a negative ratio makes the denominator larger.

State the condition on the common ratio r for an infinite geometric series to have a finite sum.

• |r| < 1
• r < 1
• −1 ≤ r ≤ 1
• 0 < r < 1

Answer: |r| < 1

Convergence requires |r| < 1, i.e. −1 < r < 1. Negative ratios are allowed, but r = 1 and r = −1 both diverge.

The recurring decimal 0.777… can be written as the series 0.7 + 0.07 + 0.007 + … Determine its value as a fraction.

• 7/10
• 7/9
• 7/11
• 70/99

Answer: 7/9

This is geometric with u₁ = 0.7 and r = 0.1, so S∞ = 0.7/(1 − 0.1) = 0.7/0.9 = 7/9.

Calculate the sum to infinity of the series 18 − 6 + 2 − 2/3 + …

• 27/2
• 27
• 9
• 18

Answer: 27/2

Here u₁ = 18 and r = −6/18 = −1/3, so S∞ = 18/(1 + 1/3) = 18/(4/3) = 27/2. Using 1 − 1/3 in the denominator (ignoring the sign of r) wrongly gives 27.

Determine the value of the infinite sum 3 + 6/5 + 12/25 + …

• 3
• 15
• 7.5
• 5

Answer: 5

The common ratio is (6/5)/3 = 2/5, so S∞ = 3/(1 − 2/5) = 3/(3/5) = 5.

An infinite geometric series has first term 4 and sum to infinity 6. Calculate its second term.

• 1/3
• 2/3
• 4/3
• 8/3

Answer: 4/3

4/(1 − r) = 6 gives r = 1/3, so the second term is u₂ = u₁r = 4 × 1/3 = 4/3.