Sum of infinite geometric sequences practice questions.

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

• 18
• 16
• 36
• 9

Answer: 18

S∞ = a/(1 − r) = 12/(1 − 1/3) = 12/(2/3) = 18.

For which value of the common ratio r does an infinite geometric series converge?

• r = 1
• |r| < 1
• |r| > 1
• r ≥ 1

Answer: |r| < 1

An infinite geometric series converges to a finite sum only when |r| < 1, so the terms shrink toward zero.

An infinite geometric series has sum to infinity 40 and first term 10. Determine the common ratio.

• 3/4
• 1/4
• 1/2
• 4

Answer: 3/4

S∞ = a/(1 − r) gives 40 = 10/(1 − r), so 1 − r = 1/4 and r = 3/4.

Calculate the sum to infinity of the series 9 − 3 + 1 − 1/3 + …

• 27/4
• 27/2
• 6
• 9/2

Answer: 27/4

Here a = 9 and r = −1/3, so S∞ = 9/(1 − (−1/3)) = 9/(4/3) = 27/4.

An infinite geometric series has sum to infinity 50 and common ratio 0.6. Determine the first term.

• 20
• 30
• 80
• 25

Answer: 20

a = S∞(1 − r) = 50 × (1 − 0.6) = 50 × 0.4 = 20.

The sum to infinity of a geometric series is three times its first term. Calculate the common ratio.

• 2/3
• 1/3
• 3
• 1/2

Answer: 2/3

a/(1 − r) = 3a gives 1/(1 − r) = 3, so 1 − r = 1/3 and r = 2/3.