We read your photo as:

Given the sum of the first n terms of an arithmetic progression (A.P.) is S_n = n2 − 8n.

Mathematics

ScanSolve AI Answer

Answer

70\textbf{70}

Step-by-step solution

6 steps
  1. Find the formula for the (n−1)th(n-1)^{th} sum, Sn−1S_{n-1}.

    Given the sum of the first nn terms of an arithmetic progression (A.P.) is Sn=n2−8nS_n = n^2 - 8n.
    To find Sn−1S_{n-1}, substitute (n−1)(n-1) for nn:

    Sn−1=(n−1)2−8(n−1)S_{n-1} = (n-1)^2 - 8(n-1)
  2. Expand and simplify Sn−1S_{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
  3. Find the formula for the nthn^{th} term, ana_n.

    The nthn^{th} term of an A.P. can be found using the formula an=Sn−Sn−1a_n = S_n - S_{n-1}.
    Substitute the expressions for SnS_n and Sn−1S_{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
  4. Set up the inequality based on the problem statement.

    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
  5. 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
  6. Determine the least possible integer value of nn.

    Since nn represents the term number, it must be an integer. The least integer value of nn that is greater than 69.5 is 70.

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

Was this helpful?

Try one yourself

A new problem that tests the same skill.

Still stuck on this one?

Ask a follow-up, and the answer starts from this question.

Got a different question?Ask your own question
Handwritten step-by-step solution preview

Handwritten Step-by-Step Solution

Get this as a handwritten image on WhatsApp — like a tutor wrote it for you.

Try on WhatsApp