If the nth Partial sum of a series A_n is S_n = n - 5/n + 5. Find A_n and A₁. 

Solution :

Given:

Nth partial sum of a series A_n is : S_n = (n - 5)/n + 5

We know that A_n = S_n - S_{n - 1}

= (n - 5)/(n + 5) - [(n - 1) - 5/(n - 1) + 5]

= (n - 5)/(n + 5) - [(n - 6)/(n + 4)]

= [{n - 5}(n + 4) - (n + 5){n - 6}] / (n + 4)(n + 5)

= {n² + 4n - 5n - 20 - (n² + 5n - 6n - 30)/ (n + 5)(n + 4)

= {n² + 4n - 5n - 20 - n² - 5n + 6n + 30)/ (n + 5)(n + 4)

= 10/ (n + 5)(n + 4)

A_n = 10/ (n + 5)(n + 4); n > 0

= Undefined; n = -4, -5

A₁ = 10/ (n + 5)(n + 4) when n = 1

A₁ = 10/(1 + 5)(1 + 4) = 10/30 = 1/3

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If the nth Partial sum of a series A_n is S_n = n - 5/n + 5. Find An and A1. 

Summary :

If the nth partial sum of a series A_n is S_n = n - 5/n + 5, then A_n is 10/(n + 5)(n + 4) ; n > 0 and, the value of A₁ is 1/3.