High School

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------------------------------------------------ 7. The nth term of an arithmetic progression (AP) is T_n = (2n + 1)^2 - 3. Determine the 10th term and the sum of the first 10 terms of the AP.

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:


  1. 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.


  2. 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]


  3. 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.