How many terms of the AP \(9, 17, 25, \ldots\) must be taken to give a sum of 636?
12.
By putting \(a = 9\), \(d = 8\), \(S = 636\) in the formula \(S = \dfrac{n}{2}[2a + (n - 1)d]\), we get the quadratic equation \(4n^2 + 5n - 636 = 0\).
On solving, we get \(n = -\dfrac{53}{4}, 12\). Out of these two roots only one root \(12\) is admissible.
Step 1: Identify \(a\), \(d\) and \(S_n\).
The AP is \(9, 17, 25, \ldots\).
First term: \(a = 9\)
Common difference: \(d = 17 - 9 = 8\)
Required sum: \(S_n = 636\)
Step 2: Use the sum formula for an AP.
For an AP, the sum of first \(n\) terms is
\[S_n = \dfrac{n}{2}[2a + (n - 1)d].\]
Substitute \(a = 9\), \(d = 8\), \(S_n = 636\):
\[636 = \dfrac{n}{2}[2\cdot 9 + (n - 1)8].\]
Step 3: Simplify inside the bracket.
\[2a = 2 \cdot 9 = 18\]
\[(n - 1)d = 8(n - 1) = 8n - 8\]
So,
\[2a + (n - 1)d = 18 + 8n - 8 = 8n + 10.\]
Thus,
\[636 = \dfrac{n}{2}(8n + 10).\]
Step 4: Form the quadratic equation in \(n\).
Multiply both sides by 2:
\[1272 = n(8n + 10).\]
Expand the right side:
\[1272 = 8n^2 + 10n.\]
Bring all terms to one side:
\[8n^2 + 10n - 1272 = 0.\]
Divide the whole equation by 2 to simplify:
\[4n^2 + 5n - 636 = 0.\]
Step 5: Solve the quadratic equation.
Use the quadratic formula for \(4n^2 + 5n - 636 = 0\):
\[n = \dfrac{-5 \pm \sqrt{5^2 - 4\cdot 4\cdot (-636)}}{2\cdot 4}.\]
Compute the discriminant:
\[D = 25 + 4\cdot 4\cdot 636 = 25 + 10176 = 10201 = 101^2.\]
So,
\[n = \dfrac{-5 \pm 101}{8}.\]
This gives two values:
\[n = \dfrac{-5 + 101}{8} = \dfrac{96}{8} = 12,\]
\[n = \dfrac{-5 - 101}{8} = \dfrac{-106}{8} = -\dfrac{53}{4}.\]
Step 6: Choose the valid value of \(n\).
Since \(n\) is the number of terms, it must be a positive integer. The value \(-\dfrac{53}{4}\) is negative and not acceptable.
Therefore, we take \(n = 12\).
Conclusion: We must take 12 terms of the AP to get a sum of 636.