Question

1cube + 3 cube+ 5cube......+(2n-1)cube =n square (2n square -1)

Answer 1

Hi,

Here is your answer !
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Given series :

1³ + 3³ + 5³ + ... + (2n - 1)³

By Adding and subtracting
2³ + 4³ + 6³ +... + (2n)³ We get

1³ + 2³ + 3³ + ... + (2n-1)³ + (2n)³
- (2³ + 4³ + 6³ + ... + (2n)³)

= [ 2n(2n+1)/2 ]² - 2³[ 1³ + 2³ + 3³+ ... + (2n)³]

= [ n(2n + 1) ]² - 2³ [ n(n + 1)/2 ]²

= n²(2n+1)² - 2³ n²(n + 1)²/2²

= n²(2n + 1)² - 2n²(n + 1)²

= n² [ (2n + 1)² - 2(n + 1)² ]

= n² [ 4n² + 4n + 1 - 2n² - 4n - 2 ]

= n² (2n² - 1)

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★ Formula Used :

Sum of cube of first n natural number

= [ n(n + 1)/2 ]²