Question

If the \(n^{th}\) term of an A.P. is \(2n + 5\), find the sum of the first 10 terms.

• A. \(110\)
• B. \(155\)
• C. \(160\)
• D. \(165\)
• E.

Hint:

If the \(n^{th}\) term of an A.P. is given as a linear expression \(An + B\), the common difference is always \(A\). You can also find the sum using \(S_n = \frac{n}{2}[2a + (n-1)d]\).

Answer 1

The Correct Option is C

Concept: In an Arithmetic Progression (A.P.), the sum of the first \(n\) terms is given by \[ S_n = \frac{n}{2}(a + l) \] where \(a\) = first term, \(l\) = last term (\(n^{th}\) term), \(n\) = number of terms.

Step 1: Find the first term. The given \(n^{th}\) term is: \[ a_n = 2n + 5 \] For the first term (\(n = 1\)): \[ a = 2(1) + 5 = 7 \]

Step 2: Find the 10th term. For the last term (\(n = 10\)): \[ l = a_{10} = 2(10) + 5 = 25 \]

Step 3: Calculate the sum of the first 10 terms. Using the formula for \(S_{10}\): \[ S_{10} = \frac{10}{2}(7 + 25) \] \[ S_{10} = 5(32) = 160 \] Hence, the sum is \[ \boxed{160} \]