the sum of the n termsof an ap whose first term is 5 and common difference is 36 is equal to the sum of 2n terms of another AP whose first term is 36 ans common difference is 5. Find n?

sum of n terms of 1st = (n/2)(10 + 36(n-1))

= (n/2)(36n - 26)
sum of 2n terms of 2nd = (2n/2)(72 + 5(2n-1))
= n(10n + 67)

(n/2)(36n - 26) = n(10n + 67)
divide by n, and multiply by 2
36n - 26 = 20n + 134
16n = 160
n = 10

check:
sum(10 for 1st) = 5(10 + 36(9)) = 1670
sum(20 for 2nd) = 10(72 + 19(5)) = 1670
all is good!

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All well

nice one mathematician

thanks

really helped........thanks

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Nice it's clear my doubts

To find the value of 'n' in this problem, we need to set up equations based on the given information and solve for 'n'.

Let's start by finding the sum of the 'n' terms of the first arithmetic progression (AP).

The formula for the sum of the first 'n' terms of an AP is: Sn = (n/2)(2a + (n-1)d), where 'a' is the first term and 'd' is the common difference.

In this case, the first term 'a' of the first AP is 5, and the common difference 'd' is 36. So, the sum of the 'n' terms of the first AP is:

Sn = (n/2)(2 * 5 + (n-1) * 36)

Next, we need to find the sum of the 2n terms of the second arithmetic progression (AP).

Using the same formula, the sum of the 2n terms of the second AP is:

S2n = (2n/2)(2 * 36 + (2n-1) * 5)

Now we can set up an equation based on the given information:

Sn = S2n

(n/2)(2 * 5 + (n-1) * 36) = (2n/2)(2 * 36 + (2n-1) * 5)

Simplifying this equation:

(n/2)(10 + 36n - 36) = ((2n)/2)(72 + 10n - 5)

(n/2)(36n - 26) = n(36n + 5n + 134)

36n^2 - 26n = 98n^2 + 5n^2 + 134n

37n^2 + 160n = 0

We can factor out 'n' from this equation:

n(37n + 160) = 0

This equation holds true if either n = 0 or 37n + 160 = 0.

But since n represents the number of terms in the AP, it cannot be zero. So we set up the equation for the non-zero value of n:

37n + 160 = 0

Solving for 'n':

37n = -160

n = -160 / 37

Therefore, the value of 'n' is approximately -4.324, which is not a feasible solution in this context because 'n' represents the number of terms and cannot be negative or fractional.

Thus, there is no valid solution for 'n' in this problem.

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