An arithmetic sequence is defined as follows:
\[\begin{cases}
a_1 = -29
\\
a_i = a_{i - 1} + 2
\end{cases}\]
Find the sum of the first
\[48\] terms in the sequence.
Answer (1)
Answer: The formula for the nth term of an arithmetic sequence is:aₙ = a₁ + (n - 1)dThe formula for the sum of the first n terms of an arithmetic sequence is:Sₙ = n/2 *[2a₁ + (n - 1)d] or equivalently Sₙ = n/2 * (a₁ + aₙ)1. Find the 48th term (a₄₈):Using the formula aₙ = a₁ + (n - 1)d with n = 48, a₁ = -29, and d = 2:a₄₈ = -29 + (48 - 1)(2) = -29 + 47(2) = -29 + 94 = 652. Find the sum of the first 48 terms (S₄₈):Using the formula Sₙ = n/2 * (a₁ + aₙ) with n = 48, a₁ = -29, and a₄₈ = 65:S₄₈ = 48/2 * (-29 + 65) = 24 * 36 = 864Therefore, the sum of the first 48 terms in the sequence is \(\boxed{864}\).Step-by-step explanation:
