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

Analyzing an A.P. given its nth term:

\[ a_n = 5 + 2n \]

Initial 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: In linear expressions for \( a_n \), the common difference is the coefficient of \( n \).

Sum of the First 20 Terms

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

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

Is 52 a Term?

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

Since \( n \) isn't a positive integer, 52 is not a term in this A.P.

Summary:

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