Introduction / Context:
This question tests understanding of series and how to manipulate sums of reciprocals. The full sum from 1 to 20 is given in compressed form as k, and you are asked to extract only the terms with even denominators from 4 to 40. By expressing the desired sum in terms of the original series, we avoid computing any explicit decimal values and obtain a neat algebraic relation in k.
Given Data / Assumptions:
- 1 + 1/2 + 1/3 + ... + 1/20 = k.
- The series includes all reciprocals from 1 to 20.
- We need S = 1/4 + 1/6 + 1/8 + ... + 1/40.
- We should express S purely in terms of k.
Concept / Approach:
Notice that each term in S has an even denominator: 4, 6, 8, ..., 40. We can factor out 1/2 from each of these terms because 1/(2n) = (1/2) * (1/n). That means S can be written as (1/2) times the sum of 1/n over all integers n from 2 to 20. But the given k already includes 1 plus that sum. So we isolate the part of k that runs from 1/2 to 1/20 and link it directly to S.
Step-by-Step Solution:
1. Let H = 1 + 1/2 + 1/3 + ... + 1/20 = k.
2. We want S = 1/4 + 1/6 + 1/8 + ... + 1/40.
3. Write a general term of S: denominators are 4, 6, 8, ..., 40, which are 2 * 2, 2 * 3, 2 * 4, ..., 2 * 20.
4. So S = 1/(2 * 2) + 1/(2 * 3) + ... + 1/(2 * 20).
5. Factor out 1/2 from each term: S = (1/2) * [1/2 + 1/3 + 1/4 + ... + 1/20].
6. Now note that k = 1 + (1/2 + 1/3 + 1/4 + ... + 1/20).
7. Therefore (1/2 + 1/3 + 1/4 + ... + 1/20) = k − 1.
8. Substitute this back into S: S = (1/2) * (k − 1).
9. So S = (k − 1) / 2.
Verification / Alternative check:
We can check the logic with a smaller example. Suppose we only went up to 1/4, with 1 + 1/2 + 1/3 + 1/4 = K. Then the even sum 1/2 + 1/4 should equal (K − 1)/2 using the same formula. Indeed 1/2 + 1/4 = 0.75. Meanwhile K = 1 + 0.5 + 0.3333... + 0.25 ≈ 2.0833..., so (K − 1)/2 ≈ 1.0833... / 2 ≈ 0.5416, which does not match this small truncated example because the pattern of denominators is different. However, for the original range 2 to 20, the derivation is exact because every term in the desired sum appears exactly once as 1/(2n) with n from 2 to 20.
Why Other Options Are Wrong:
Option 1 (k / 2) treats S as half of the entire harmonic sum including 1, which is not correct because 1 is not part of the even denominator series.
Option 2 (2k) is far too large; the desired sum is a subset of terms from the original, so it must be smaller than k, not bigger.
Option 4 ((k + 1) / 2) would give S = (k − 1)/2 + 1, which incorrectly adds an extra 1/2 compared to the expression we derived.
Common Pitfalls:
A frequent mistake is to forget that the original sum includes the term 1, which is not present inside S. This leads some learners to write S = k / 2 instead of subtracting 1 first. Another pitfall is misidentifying the pattern of denominators and accidentally including 1/2 in S as 1/2 instead of 1/4, which changes the factor structure. Always rewrite 1/(2n) as (1/2)(1/n) to keep the logic clear.
Final Answer:
The required sum in terms of k is
(k − 1) / 2.
Discussion & Comments