Question

Let S_n be the sum of the first n terms of the series a₁+a₂+...a_n+… If S_n=n²+4n, then the n^th term a_n is

• A. 2n+3
• B. 2n-1
• C. 2n+5
• D. 2n-3
• E. 2n

Answer 1

The Correct Option is A

We are given the sum of the first \( n \) terms of the series:

\[ S_n = n^2 + 4n \] The \( n \)-th term \( a_n \) of the series is given by: \[ a_n = S_n - S_{n-1} \]

Step 1: Find \( S_n \) and \( S_{n-1} \)

From the given formula for \( S_n \), we have: \[ S_n = n^2 + 4n \] Now, substitute \( n-1 \) into the formula for \( S_n \) to find \( S_{n-1} \): \[ S_{n-1} = (n-1)^2 + 4(n-1) \] Expanding this: \[ S_{n-1} = (n^2 - 2n + 1) + 4n - 4 = n^2 + 2n - 3 \]

Step 2: Find \( a_n \)

Now, subtract \( S_{n-1} \) from \( S_n \) to find \( a_n \): \[ a_n = S_n - S_{n-1} = (n^2 + 4n) - (n^2 + 2n - 3) \] Simplify the expression: \[ a_n = n^2 + 4n - n^2 - 2n + 3 = 2n + 3 \] Thus, the \( n \)-th term is: \[ a_n = 2n + 3 \]

The correct option is (A) : \(2n+3\)

Answer 2

We are given that the sum of the first \(n\) terms, \(S_n\), is given by:

\[ S_n = n^2 + 4n \]

The nth term of the series, \(a_n\), is the difference between the sum of the first \(n\) terms and the sum of the first \(n-1\) terms:

\[ a_n = S_n - S_{n-1} \]

Substitute the given formula for \(S_n\) and \(S_{n-1}\):

\[ S_n = n^2 + 4n, \quad S_{n-1} = (n-1)^2 + 4(n-1) \]

Now calculate \(S_{n-1}\):

\[ S_{n-1} = (n-1)^2 + 4(n-1) = n^2 - 2n + 1 + 4n - 4 = n^2 + 2n - 3 \]

Now, find \(a_n\):

\[ a_n = S_n - S_{n-1} = (n^2 + 4n) - (n^2 + 2n - 3) \] \[ a_n = n^2 + 4n - n^2 - 2n + 3 = 2n + 3 \]

Thus, the nth term of the series is 2n + 3.