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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 n for which the nth term is greater than 130.

Mathematics

ScanSolve AI Answer

Answer

70\text{70}

Step-by-step solution

5 steps
  1. Find the formula for the nthn^{th} term, ana_n.

    The sum of the first nn terms of an arithmetic progression is given by Sn=n2−8nS_n = n^2 - 8n.
    The nthn^{th} term, ana_n, can be found using the relationship an=Sn−Sn−1a_n = S_n - S_{n-1}.

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

    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
  2. Calculate ana_n.

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

    an=Sn−Sn−1a_n = S_n - S_{n-1} an=(n2−8n)−(n2−10n+9)a_n = (n^2 - 8n) - (n^2 - 10n + 9) an=n2−8n−n2+10n−9a_n = n^2 - 8n - n^2 + 10n - 9 an=2n−9a_n = 2n - 9
  3. Set up the inequality.

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

    an>130a_n > 130

    Substitute the expression for ana_n:

    2n−9>1302n - 9 > 130
  4. 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
  5. Determine the least possible integer value of nn.

    Since nn must be an integer (as it represents the term number) 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{\text{70}}.

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