Answer :
Let's solve this question step by step:
1. Understand the nth term of the AP:
- We are given that the nth term, [tex]\( T_n \)[/tex], of an arithmetic progression (AP) is [tex]\( 2n + 1 \)[/tex].
2. Find the first term:
- The first term of the AP, when [tex]\( n = 1 \)[/tex], is found by substituting [tex]\( n = 1 \)[/tex] into the formula for the nth term:
[tex]\[
T_1 = 2 \times 1 + 1 = 3
\][/tex]
3. Determine the sum of the first n terms:
- The formula for the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of an AP is given by:
[tex]\[
S_n = \frac{n}{2} \times (\text{first term} + \text{last term})
\][/tex]
- The first term is [tex]\( 3 \)[/tex], which we have already found.
- The last term, or the nth term, is [tex]\( T_n = 2n + 1 \)[/tex].
4. Substitute the known values into the sum formula:
- Substitute the first term and the nth term into the sum formula:
[tex]\[
S_n = \frac{n}{2} \times (3 + (2n + 1))
\][/tex]
- Simplify the expression inside the parentheses:
[tex]\[
S_n = \frac{n}{2} \times (2n + 4)
\][/tex]
- Further simplifying, we factor out a 2 from the expression:
[tex]\[
S_n = \frac{n}{2} \times 2 \times (n + 2)
\][/tex]
- Cancel the 2 in the numerator and denominator:
[tex]\[
S_n = n \times (n + 2)
\][/tex]
Therefore, the sum of the first [tex]\( n \)[/tex] terms of the AP is [tex]\( n(n + 2) \)[/tex].
1. Understand the nth term of the AP:
- We are given that the nth term, [tex]\( T_n \)[/tex], of an arithmetic progression (AP) is [tex]\( 2n + 1 \)[/tex].
2. Find the first term:
- The first term of the AP, when [tex]\( n = 1 \)[/tex], is found by substituting [tex]\( n = 1 \)[/tex] into the formula for the nth term:
[tex]\[
T_1 = 2 \times 1 + 1 = 3
\][/tex]
3. Determine the sum of the first n terms:
- The formula for the sum [tex]\( S_n \)[/tex] of the first [tex]\( n \)[/tex] terms of an AP is given by:
[tex]\[
S_n = \frac{n}{2} \times (\text{first term} + \text{last term})
\][/tex]
- The first term is [tex]\( 3 \)[/tex], which we have already found.
- The last term, or the nth term, is [tex]\( T_n = 2n + 1 \)[/tex].
4. Substitute the known values into the sum formula:
- Substitute the first term and the nth term into the sum formula:
[tex]\[
S_n = \frac{n}{2} \times (3 + (2n + 1))
\][/tex]
- Simplify the expression inside the parentheses:
[tex]\[
S_n = \frac{n}{2} \times (2n + 4)
\][/tex]
- Further simplifying, we factor out a 2 from the expression:
[tex]\[
S_n = \frac{n}{2} \times 2 \times (n + 2)
\][/tex]
- Cancel the 2 in the numerator and denominator:
[tex]\[
S_n = n \times (n + 2)
\][/tex]
Therefore, the sum of the first [tex]\( n \)[/tex] terms of the AP is [tex]\( n(n + 2) \)[/tex].
