The sum to infinity of a geometric sequence is 20 and the common ratio r is 0.2. What are the first four terms of the sequence?

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BlackTom AI · Accepted Answer

Let’s start by identifying the key information: the sum to infinity of a geometric sequence is 20, and the common ratio r is 0.2. For a geometric series with |r| < 1, the sum to infinity S∞ is given by S∞ = a1 / (1 - r), where a1 is the first term. Using S∞ = 20 and r = 0.2, we can solve for a1: a1 = S∞ * (1 - r) = 20 * 0.8 = 16. Now we can generate the first few terms by multiplying by r successively: a2 = a1 * r = 16 * 0.2 = 3.2, a3 = a2 * r = 3.2 * 0.2 = 0.64, a4 = a3 * r = 0.64 * 0.2 = 0.128.

Option A: 18, 3.6, 0.72, 0.144
- This sequence would imply a1 = 18, which would give a sum to infinity of 18 / (1 - 0.2) = 18 / 0.8 = 22.5, not 20. Therefore this option is inconsistent with the given sum.

Option B: 15, 3, 0.6, 0.12
- Here a1 appears to be 15, leading to S∞ = 15 / 0.8 = 18.75, which does not match the required sum of 20. This shows a mismatch between the stated first term and the given sum.

Option C: 16, 3.2, 0.64, 0.128
- This set starts at 16, and each term is 0.2 times the previous one, matching the calculated a1 = 16 and the subsequent terms 3.2, 0.64, 0.128. If you recompute the sum to infinity using a1 = 16 and r = 0.2, you indeed get 16 / (1 - 0.2) = 16 / 0.8 = 20, which aligns with the given information. This is internally consistent.

Option D: 20, 4, 0.8, 0.16
- This suggests a1 = 20, which would yield S∞ = 20 / 0.8 = 25, not 20. Additionally, the subsequent term would be a2 = 4, which is 0.2 times 20, but the calculated a1 already shows inconsistency with the specified sum to infinity.

In summary, each option is tested for internal consistency with the given sum and ratio. Options A, B, and D fail to satisfy the sum-to-infinity condition with r = 0.2, while Option C satisfies both the first-term derivation from the sum formula and the successive term multiplication by r.

The sum to infinity of a geometric sequence is 20 and the common ratio r is 0.2. What are the first four terms of the sequence?

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