8, 19, 30, 41, 52, …. Given the sequence above, what is the sum of the 10th and the 20th terms?

A. 324 B. 335 C. 346 D. 357 E. 368

(A) First, we need to derive a formula so that we do not need to write out the first 20 terms. The first term is 8 = 11 – 3. The second term is 19 = 22 – 3, the third term is 30 = 3 × 11 – 3, the fourth term is 41 = 4 × 11 – 3, etc. Thus, 11n – 3 describes the values in the sequence where n is the number of the term. The 10th term is 10 × 11 – 3 = 107. The 20th term is 20 × 11 – 3 = 220 – 3 = 217. The sum of these two values is 324. The correct answer is choice (A). ---------- Can you please tell me how you came up with the formula: 8 = (11 – 3). 8 + 11 = 19, 19 + 11 = 30, etc.

If a sequence is given and you're asked to find the sequence, you should look for the "simple" sequences first, like arithmetic or geometric progressions.

Does the sequence always start at one as opposed to 0?

A sequence starts with the first element. If you think of the first element as of the zero one, then you must shift all other elements accordingly. So if a question asks you to calculate the sum of the 10th and the 20th terms, then in your notation it will be the sum of the 9th and 19th terms.

Student wrote:

I used Sn = 11n + 8 and got the trap answer of 346. Please advise.

If you notate the first element as S0 (S0 = 8), then the 10th one will be S9 and the 20th one will be S19.

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