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in Arithmetic Progression by (56.3k points)

The first term of an A.P. is 5, the last term is 45 and the sum is 400. Find the number of terms and the common difference.

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Sum of first n terms of an A.P is given by Sn = \(\frac{n}{2}\)(2a + (n − 1)d)

Given,

First term (a) = 5, last term (an) = 45 and sum of n terms (Sn) = 400

Now, we know that

an = a + (n – 1)d

⟹ 45 = 5 + (n – 1)d

⟹ 40 = nd – d

⟹ nd – d = 40 …. (1)

Also,

S= \(\frac{n}{2}\)(2(a) + (n − 1)d)

400 = \(\frac{n}{2}\)(2(5) + (n − 1)d)

800 = n (10 + nd – d)

800 = n (10 + 40) [using (1)]

⟹ n = 16

Putting n in (1), we find d

nd – d = 40

16d – d = 40

15d = 40

d = \(\frac{8}{3}\)

Therefore, the common difference of the given A.P. is \(\frac{8}{3}\).

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