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1 + \frac { 4 } { 5 } + \frac { 7 } { 25 } + \frac { 10 } { 125 } + |
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Answer» i) Given: S = 1 + 4/5 + 7/25 + 10/125 + 13/625 + .... ---------- (1) ii) Multiplying the above by 1/5 both sides[Why 1/5? Reason: In the given sequence every term on right side is multiplied by 1/5] S/5 = 1/5 + 4/25 + 7/125 + 10/625 + ..... ==> S/5 = 0 + 1/5 + 4/25 + 7/125 + 10/625 + ...... ------ (2) iii) (1) - (2): Subtract the corresponding terms: (S - S/5) = (1 - 0) + (4/5 - 1/5) + (7/25 - 4/25) + (13/125 - 10/125) + .... ==> 4S/5 = 1 + 3/5 + 3/25 + 3/125 + ..... ==> 4S/5 = 1 + (3/5)(1 + 1/5 + 1/25 + 1/125 + ...) ! + 1/5 + 1/25 + 1/125 + .... is a infinite GP, with first term = 1 and common ratio = 1/5,whose sum to infinite terms = a/(1 - r) = 1/(1 - 1/5) = 5/4 Thus, 4S/5 = 1 + (3/5)(5/4) = 1 + 3/4 = 7/4 Hence S = (7/4)*(5/4) = 35/16 So sum of the given sequence to infinite terms = 35/16 |
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