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Sum

The first term of an AP is 3, the last term is 83 and the sum of all its terms is 903. Find the number of terms and the common difference of the AP.

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#### Solution

**Given:**

a = 3

l = 83

Sum all the n terms = 903

S_{n} = 903

`("n")/(2) [ "a" + "l"] = 903`

`("n")/(2). (3+83) = 903`

`"n". 43 = 903`

`"n" = (903)/(43)`

n = 21

Number of terms = 21

∵l = 83

a + (n - 1) d = 83

3 + (21 - 1) . d = 83

20d = 80

d = 4

∴ Common difference = 4

∴ n = 21 and d = 4

Concept: Algebraic Conditions for Number of Solutions

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