Answer :
To find the sum of the first 10 terms of an arithmetic sequence, use the formula S_n = n/2(2a + (n - 1)d). For the sequence 69, 131, 193, with a common difference of 62, the sum of the first 10 terms (S_<10>) is 3480.
To find the sum of the first 10 terms (S_<10>) of an arithmetic sequence, we utilize the formula for the sum of an arithmetic series, which is S_n = n/2(2a + (n - 1)d), where 'n' is the number of terms, 'a' is the first term, and 'd' is the common difference between the terms.
The given arithmetic sequence is 69, 131, 193, ...
First, we need to identify the common difference 'd'.
By subtracting the first term from the second term, we find that d = 131 - 69 = 62.
The first term 'a' is given as 69, and we already know that we are looking for the first 10 terms, so n = 10.
Substituting these values into the sum formula results in: S_<10> = 10/2(2 * 69 + (10 - 1) * 62) = 5(138 + 9 * 62)= 5(138 + 558)= 5(696)= 3480
Therefore, the sum of the first 10 terms of the given arithmetic sequence is 3480. When solving such problems, always ensure that you have the correct values for 'a', 'n', and 'd' to give final answer accurately.
