find-sum-of-the-series-1-2-3-2-5-2-7-2-9-2-11-2-4n-3-2-4n-1-2-

Question Number 114433 by bemath last updated on 19/Sep/20

f i n d s u m o f t h e s e r i e s

1^{2} - 3^{2} + 5^{2} - 7^{2} + 9^{2} - 11^{2} + \dots + {(4 n - 3)}^{2} - {(4 n - 1)}^{2}

Answered by 1549442205PVT last updated on 19/Sep/20

S = 1 + (5^{2} - 3^{2}) + (9^{2} - 7^{2}) + (13^{2} - 11^{2}) + \dots [{(4 n - 3)}^{2} - {(4 n - 5)}^{2}] - {(4 n - 1)}^{2}

= 1 + (5 + 3) (5 - 3) + (9 + 7) (9 - 7) + (13 + 11) (13 - 11) + \dots + (8 n - 8) . 2 - {(4 n - 1)}^{2}

= 1 + 2 [8 + 16 + 24 + \dots + (8 n - 8)] - {(4 n - 1)}^{2}

= 1 + 2 [\frac{(8 + 8 n - 8) (n - 1)}{2}] - {(4 n - 1)}^{2}

= 1 + 8 n (n - 1) - {16 n}^{2} + 8 n - 1

S = - {8 n}^{2}

Commented by bemath last updated on 19/Sep/20

y e s . . g a v e k u d o s ✓

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