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 and the common difference . We need to find how many terms, denoted as , must be taken so that the sum of the terms is 636.
The sum of the first terms of an AP is given by the formula:
Substituting the known values:
Simplifying:
Now, we'll solve this quadratic equation using the quadratic formula:
Here, , , and . Plugging in these values:
Now, we have two possible solutions:
Since must be a positive integer, the solution is .
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:
- What is the formula for the sum of an arithmetic progression?
- How do you solve a quadratic equation using the quadratic formula?
- What is the general term of an arithmetic sequence?
- How do you find the common difference in an AP?
- 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
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