How many term of arithmetic progression,9, 17,25.are to be taken so their sum is 636?

Question

Answer

A.P = 9, 17, 25 A= 9 Common difference =17-9=8 sum of n terms= n÷2[2a+(n-1)d 636=n÷2[2(9)+(n-1)8] 636= n÷2[18+8n-8] 636=n÷2[10+8] 1272=10n+8n^2 8n^2+10n-1272=0 2[4n^2+5n-636]=0 4n^2+5n-636=0 4n^2+53n-48n -636=0 n(4n+53)-12(4n+53)=0 (n-12)(4n+53)=0 n-12=0 n=12 12 terms must be given to sum of 636..

Answer

How many terms of arithmetic progression 63,60,57,.... are to be taken so their sum is 693

Answer

Arithmetic Progression:
An arithmetic progression is a sequence of numbers in which the difference between any two consecutive terms is constant. The first term is denoted by 'a' and the common difference by 'd'.

Finding the Number of Terms:
We are given the arithmetic progression 9, 17, 25 and we need to determine how many terms should be taken in order for their sum to be 636.

Step 1: Find the Common Difference:
To find the common difference, we subtract the first term from the second term:
17 - 9 = 8

Step 2: Find the Last Term:
We can find the last term of the arithmetic progression by using the formula:
Last Term (L) = a + (n - 1)d

In this case, the first term (a) is 9 and the common difference (d) is 8. We need to find the value of 'n' (number of terms) when the sum is 636.

Step 3: Substitute the Values:
L = 9 + (n - 1)8

Simplifying the equation:
L = 9 + 8n - 8
L = 1 + 8n

Step 4: Find the Sum:
The sum of an arithmetic progression can be calculated using the formula:
Sum (S) = (n/2)(a + L)

Substituting the values:
636 = (n/2)(9 + 1 + 8n)

Simplifying the equation:
636 = (n/2)(10 + 8n)
1272 = n(10 + 8n)

Simplifying further:
1272 = 10n + 8n^2

Step 5: Solve the Quadratic Equation:
To find the value of 'n', we need to solve the quadratic equation:
8n^2 + 10n - 1272 = 0

Using the quadratic formula:
n = (-b ± √(b^2 - 4ac)) / 2a

Substituting the values:
n = (-10 ± √(10^2 - 4(8)(-1272))) / 2(8)

Calculating the discriminant:
n = (-10 ± √(100 + 40704)) / 16
n = (-10 ± √40804) / 16
n = (-10 ± 202) / 16

Simplifying further:
n = (192/16) or n = (-212/16)

Step 6: Determine the Number of Terms:
We discard the negative value as the number of terms cannot be negative. Hence, we take the positive value of 'n':
n = (192/16)
n = 12

Therefore, to have a sum of 636, we need to take 12 terms in the arithmetic progression 9, 17, 25.

Similar Class 10 Doubts

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