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The sum of the first n terms of an arithmetic progression is given by S_n = n2 − 8n. Find the least integer value of n for which the nth term is greater than 130.

Mathematics

ScanSolve AI Answer

Answer

n=70\mathbf{n=70}

Step-by-step solution

4 steps
  1. Find the formula for the nthn^{th} term, TnT_n, using the given sum SnS_n.

    The sum of the first nn terms of an A.P. is given by Sn=n2−8nS_n = n^2 - 8n.
    The nthn^{th} term, TnT_n, can be found using the formula Tn=Sn−Sn−1T_n = S_n - S_{n-1}.

    First, find Sn−1S_{n-1} by replacing nn with (n−1)(n-1) in the SnS_n formula:

    Sn−1=(n−1)2−8(n−1)S_{n-1} = (n-1)^2 - 8(n-1)

    Expand the expression:

    Sn−1=(n2−2n+1)−(8n−8)S_{n-1} = (n^2 - 2n + 1) - (8n - 8) Sn−1=n2−2n+1−8n+8S_{n-1} = n^2 - 2n + 1 - 8n + 8 Sn−1=n2−10n+9S_{n-1} = n^2 - 10n + 9

    Now, substitute SnS_n and Sn−1S_{n-1} into the formula for TnT_n:

    Tn=(n2−8n)−(n2−10n+9)T_n = (n^2 - 8n) - (n^2 - 10n + 9) Tn=n2−8n−n2+10n−9T_n = n^2 - 8n - n^2 + 10n - 9 Tn=2n−9T_n = 2n - 9
  2. Set up the inequality based on the given condition.

    The problem states that the nthn^{th} term of the sequence is greater than 130.

    Tn>130T_n > 130

    Substitute the expression for TnT_n:

    2n−9>1302n - 9 > 130
  3. Solve the inequality for nn.

    Add 9 to both sides:

    2n>130+92n > 130 + 9 2n>1392n > 139

    Divide by 2:

    n>1392n > \frac{139}{2} n>69.5n > 69.5
  4. Determine the least possible integer value of nn.

    Since nn must be an integer (representing the term number), the least integer value of nn that is greater than 69.5 is 70.

    The final answer is n=70\mathbf{n=70}.

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