Question

If the nth term of an AP is 2 n+1, find the sum of first n terms of the AP​

Answer 1

Answer :

➥ The sum of first n terms of the AP = n² + 2n

Given :

➤ aₙ = 2n + 1

To Find :

➤ Sum of first n terms of the AP = ?

Solution :

◈ a₁ = (2 × 1 + 1) = 3

◈ a₂ = (2 × 2 + 1) = 5

∴ a = 3

⟾ d = a₁ - a₂

⟾ d = 5 - 3

⟾ d = 2

Now ,

$\tt{: \implies s_{n} = \dfrac{n}{2} \left\{ 2a + (n - 1)d \right \}}$

$\tt{: \implies s_{n} = \dfrac{n}{2} \{ 2 \times 3(n - 1) \times 2\} }$

$\tt{: \implies s_{n} = \dfrac{n}{2}(6 + 2n - 2) }$

$\tt{: \implies s_{n} = \dfrac{n}{2} (6 - 2 + 2n)}$

$\tt{: \implies s_{n} = \dfrac{n}{2} (4 + 2n)}$

$\tt{: \implies s_{n} = n(n + 2)}$

$\bf{: \implies \underline{ \: \: \underline{ \purple{ \: \: s_{n} = {n}^{2} + 2n \: \: }} \: \: }}$

Hence, the sum of first n terms of the AP is n² + 2n.