Algebra problems questions

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Find the sum to infinity of the series 1² + 2². x + 3².x² + 4².x³ +…,…. |x| < 1.

Solution: Though the given series is not an Arithmetico-Geometric series, however, the differences (2² – 1²), (3² – 2²) , …..form an AP. So, we can use the Method of Differences.

Let S = 1 + 4x + 9x² +16x³ +…. ∞. 
Multiply both sides with common ratio x of the GP.
Sx = x + 4x² + 9x³ +……∞. 
Now, subtract the two equations.
=> (1 – x) S = 1 + 3x + 5x² + 7x³ +….∞ ……. (1)
Now, let R = 1 + 3x + 5x² + 7x³ +…∞, which is an Arithmetico-Geometric series with a = 1, d = 2 and r = x.

For an A.G.P, 
Sum R =  ,
Substituting the values, we get R =  ,
Substitute R in (1), we get, 
(1 – x) S = 
⇒ S = ,