Math Problem Statement

How many terms of the AP:9,17,25,..........must be taken to give a sum of 636

Solution

We are given an arithmetic progression (AP) with the first term a=9a = 9 and the common difference d=179=8d = 17 - 9 = 8. We need to find how many terms, denoted as nn, must be taken so that the sum of the terms is 636.

The sum SnS_n of the first nn terms of an AP is given by the formula:

Sn=n2×[2a+(n1)d]S_n = \frac{n}{2} \times [2a + (n-1)d]

Substituting the known values:

636=n2×[2(9)+(n1)(8)]636 = \frac{n}{2} \times [2(9) + (n-1)(8)]

Simplifying:

636=n2×[18+8n8]636 = \frac{n}{2} \times [18 + 8n - 8] 636=n2×(8n+10)636 = \frac{n}{2} \times (8n + 10) 636=n(8n+10)2636 = \frac{n(8n + 10)}{2} 1272=n(8n+10)1272 = n(8n + 10) 1272=8n2+10n1272 = 8n^2 + 10n 8n2+10n1272=08n^2 + 10n - 1272 = 0

Now, we'll solve this quadratic equation using the quadratic formula:

n=b±b24ac2an = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Here, a=8a = 8, b=10b = 10, and c=1272c = -1272. Plugging in these values:

n=10±1024(8)(1272)2(8)n = \frac{-10 \pm \sqrt{10^2 - 4(8)(-1272)}}{2(8)} n=10±100+4070416n = \frac{-10 \pm \sqrt{100 + 40704}}{16} n=10±4080416n = \frac{-10 \pm \sqrt{40804}}{16} n=10±20216n = \frac{-10 \pm 202}{16}

Now, we have two possible solutions:

n=10+20216=19216=12n = \frac{-10 + 202}{16} = \frac{192}{16} = 12 n=1020216=21216=13.25n = \frac{-10 - 202}{16} = \frac{-212}{16} = -13.25

Since nn must be a positive integer, the solution is n=12n = 12.

Therefore, 12 terms of the arithmetic progression must be taken to give a sum of 636.

Would you like more details or have any questions?

Here are 5 related questions:

  1. What is the formula for the sum of an arithmetic progression?
  2. How do you solve a quadratic equation using the quadratic formula?
  3. What is the general term of an arithmetic sequence?
  4. How do you find the common difference in an AP?
  5. Can the sum of terms in an arithmetic progression ever be negative?

Tip: In arithmetic progressions, recognizing patterns in common differences can help you quickly find general terms and sums.

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Math Problem Analysis

Mathematical Concepts

Arithmetic Progression
Quadratic Equations

Formulas

Sum of an AP: S_n = (n/2) × [2a + (n-1)d]
Quadratic equation formula: ax^2 + bx + c = 0
Quadratic formula: n = (-b ± √(b² - 4ac)) / 2a

Theorems

Quadratic Formula

Suitable Grade Level

Grades 9-11