he sum of the series 105 2 − 103 2 + 104 2 − 102 2 + 103 2 − 101 2 + 102 2 − 100 2 + … up to 40 terms is
A : | 5 520 | ||
B : | 5 560 | ||
C : | 7 500 | ||
D : | 7 560 |
plz explain this sum
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The correct solution of the question is as follows:
The given series is,
1052 – 1032 + 1042 – 1022 + 1032 – 1012 + 1022 – 1002 + ... (up to 40 terms)
The given series can be written as,
(105 – 103) (105 + 103) + (104 – 102) (104 + 102) + (103 – 101) (103 + 101) + ... (up to 20 terms) = S20 (say)
∴ S20 = (105 + 103)2 + (104 + 102)2 + (103 + 101)2 + (102 + 100)2 + ... (up to 20 terms)
= 2 [105 + 104 + 103 + 102 + ... (up to 20 terms)] + 2 [103 + 102 + 101 + 100 + ... (up to 20 terms)]
Thus, the sum of the given series is 7560.