The question "What is the sum of the arithmetic series 2, 5, 8, 20?" is unclear. Is '20' the last term of the series, or the number of terms? We will address both interpretations.
Interpretation 1: 20 is the last term.
If 20 is the last term in the arithmetic series 2, 5, 8, ..., 20, we first need to determine the number of terms.
- The common difference (d) is 5 - 2 = 3.
- The nth term of an arithmetic sequence is given by: an = a1 + (n - 1)d, where a1 is the first term and d is the common difference.
- We want to find n such that an = 20. Therefore, 20 = 2 + (n - 1)3.
- Simplifying, we get 18 = (n - 1)3, which means n - 1 = 6, and n = 7.
- Now we can find the sum of the first 7 terms using the formula: Sn = (n/2)(a1 + an)
- S7 = (7/2)(2 + 20) = (7/2)(22) = 7 * 11 = 77.
Therefore, if 20 is the last term, the sum of the series is 77.
Interpretation 2: 20 is the number of terms.
If 20 is the number of terms, we can calculate the sum of the first 20 terms of the arithmetic series.
- We still have a1 = 2 and d = 3.
- First, find the 20th term: a20 = 2 + (20 - 1)3 = 2 + (19)3 = 2 + 57 = 59.
- Then, calculate the sum using the formula: Sn = (n/2)(a1 + an)
- S20 = (20/2)(2 + 59) = 10 * 61 = 610.
Therefore, if 20 is the number of terms, the sum of the series is 610. This result aligns with the reference, which states "In the given A.P. = 610".
