What is the nth Term of the Sum of the First n Terms of an AP is Given by sn = 2n2 + 5n?

The nth term of the arithmetic progression (AP) is 4n + 3.

Here's how we arrive at this answer:

Deriving the nth Term

Given the sum of the first n terms, S_n = 2n² + 5n, we can find the nth term (a_n) using the following logic:

• S_n represents the sum of the first n terms: a₁ + a₂ + ... + a_n.
• S_n-1 represents the sum of the first n-1 terms: a₁ + a₂ + ... + a_n-1.
• Therefore, the nth term (a_n) is the difference between S_n and S_n-1: a_n = S_n - S_n-1.

Calculation

1. Find S_n-1:

   Substitute n-1 for n in the expression for S_n:

   S_n-1 = 2(n-1)² + 5(n-1)
   S_n-1 = 2(n² - 2n + 1) + 5n - 5
   S_n-1 = 2n² - 4n + 2 + 5n - 5
   S_n-1 = 2n² + n - 3

2. Calculate a_n = S_n - S_n-1:

   a_n = (2n² + 5n) - (2n² + n - 3)
   a_n = 2n² + 5n - 2n² - n + 3
   a_n = 4n + 3

Therefore, the nth term of the AP is 4n + 3.