Question

Find the sum of first 20 terms of an A.P. whose n\(^{th}\) term is given by \(a_n = 5 + 2n\). Can 52 be a term of this A.P. ?

Answer 1

Arithmetic Progression (A.P.) Analysis

Given the nth term of the A.P.:

\[ a_n = 5 + 2n \]

First Few Terms

First term: \[ a_1 = 5 + 2(1) = 7 \] Second term: \[ a_2 = 5 + 2(2) = 9 \] Common difference: \[ d = a_2 - a_1 = 9 - 7 = 2 \]

Note: For linear expressions of \( a_n \), the coefficient of \( n \) directly gives the common difference.

Sum of the First 20 Terms

The formula for the sum of the first \( n \) terms is: \[ S_n = \frac{n}{2}[2a + (n - 1)d] \]

Substituting \( n = 20 \), \( a = 7 \), and \( d = 2 \): \[ S_{20} = \frac{20}{2}[2(7) + (20 - 1)(2)] = 10[14 + 38] = 10 \times 52 = \boxed{520} \]

Can 52 Be a Term of This A.P.?

Let \( a_n = 52 \). Solve: \[ 5 + 2n = 52 \Rightarrow 2n = 47 \Rightarrow n = \frac{47}{2} = 23.5 \]

Since \( n \) must be a positive integer, 52 is not a term of this A.P.

Conclusion:

• Sum of the first 20 terms: \( \boxed{520} \)
• 52 is not a term of the A.P. since \( n = 23.5 \) is not an integer.