Answer :
To find the 10th term and the sum of the first 10 terms of the arithmetic progression (AP) given by the nth term formula [tex]T_n = (2n + 1)^2 - 3[/tex], we need to follow these steps:
Find the 10th term ([tex]T_{10}[/tex]):
Substitute [tex]n = 10[/tex] into the formula for the nth term:
[tex]T_{10} = (2 \times 10 + 1)^2 - 3[/tex]
Simplify the expression:
[tex]T_{10} = (20 + 1)^2 - 3 = 21^2 - 3[/tex]
[tex]T_{10} = 441 - 3 = 438[/tex]
Therefore, the 10th term is 438.
Determine the first term ([tex]T_1[/tex]) and the common difference (d):
Substitute [tex]n = 1[/tex] into the formula to find the first term:
[tex]T_1 = (2 \times 1 + 1)^2 - 3 = (2 + 1)^2 - 3 = 3^2 - 3[/tex]
[tex]T_1 = 9 - 3 = 6[/tex]
Now, find the second term ([tex]T_2[/tex]) to determine the common difference:
[tex]T_2 = (2 \times 2 + 1)^2 - 3 = (4 + 1)^2 - 3 = 5^2 - 3[/tex]
[tex]T_2 = 25 - 3 = 22[/tex]
Calculate the common difference (d):
[tex]d = T_2 - T_1 = 22 - 6 = 16[/tex]
Calculate the sum of the first 10 terms ([tex]S_{10}[/tex]):
The sum of the first n terms of an AP is given by the formula:
[tex]S_n = \frac{n}{2} \times (2T_1 + (n-1)d)[/tex]
Substitute [tex]n = 10[/tex], [tex]T_1 = 6[/tex], and [tex]d = 16[/tex] into the formula:
[tex]S_{10} = \frac{10}{2} \times (2 \times 6 + (10 - 1) \times 16)[/tex]
Simplify the expression:
[tex]S_{10} = 5 \times (12 + 9 \times 16) = 5 \times (12 + 144)[/tex]
[tex]S_{10} = 5 \times 156 = 780[/tex]
Therefore, the sum of the first 10 terms is 780.
In conclusion, the 10th term of the given AP is 438, and the sum of the first 10 terms is 780.
