40^3 - 36^3 +34^3 -32^3 ...+ 4^3 -2^3
Answers
Given : 40³ - 38³ + 36³ - 34³ +................ + 4³ - 2³
To find : Value
Solution:
Correction in Question :
40³ - 38³ + 36³ - 34³ +................ + 4³ - 2³
2³ ( 20³ - 19³ + 18³ - 17³ +...........2³ - 1³)
= 2³( 20³ + 18³ +...........+ 2³) - 2³(19³ + 17³ +...............+ 1³)
= 2²(2³)(10³ + 9³ + ------+ 1³) - 2³( 19³ + 17³ +...............+ 1³)
= 64 ∑n³ - 8 ∑(2n - 1)³
= 64 ∑n³ - 8 ∑(8n³ - 12n² + 6n - 1)
= 64 ∑n³ - 64 ∑n³ - 8∑(12n² + 6n - 1)
= 8∑ (12n²- 6n + 1)
= 96 ∑n² - 48∑n + 8n
n = 10
∑n² = n(n+1)(2n+1)/6 = 385
∑n = n(n +1)/2 = 55
= 96(385) - 48(55) + 80
= 46960 - 2640 + 80
= 34400
40³ - 38³ + 36³ - 34³ +................ + 4³ - 2³ = 34400
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