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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 possible value of n such that the nth term is greater than 130.

Mathematics

ScanSolve AI Answer

Answer

70

Step-by-step solution

4 steps

Here's how to find the least possible value of nn.

  1. Find the formula for the nnth term (TnT_n) of the AP.

    The sum of the first nn terms is given by Sn=n2−8nS_n = n^2 - 8n.
    The nnth term, TnT_n, can be found using the relationship Tn=Sn−Sn−1T_n = S_n - S_{n-1}.
    First, find Sn−1S_{n-1}:

    Sn−1=(n−1)2−8(n−1)S_{n-1} = (n-1)^2 - 8(n-1) 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, calculate TnT_n:

    Tn=Sn−Sn−1T_n = S_n - S_{n-1} 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 nnth term 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 of the inequality:

    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 (as it represents the term number in a sequence) and nn must be greater than 69.5, the least possible integer value for nn is 70.

    The least possible value of nn is 70\boxed{70}.

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