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 = ,