The sum of the series Sigma(r = 0)^(10) ...

The sum of the series `Sigma_(r = 0)^(10) ""^(20)C_(r) " is " 2^(19) + (""^(20)C_(10))/(2)`

Text Solution

Verified by Experts

Given series `= underset(r = 0)overset(10)Sigma .^(20)C_(r) = .^(20)C_(0) + .^(20)C_(1) + .^(20)C_(2) + ... + .^(20)C_(10)`
`= .^(20)C_(0) + .^(20)C_(1) +..... + .^(20)C_(10) + .^(20)C_(11) + ... .^(20)C_(20) - (.^(20)C_(11) + .... + .^(20)C_(20))`
`= 20^(20) - (.^(20)C_(11) + ... + .^(20)C_(20))`
Hence, the given statement is false

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The sum of the series sum_(r=0)^(10)""^(20)C_(r ) is

`2^(10)`

`2^(19)`

`2^(19) +(1)/(2)""^(20)C_(10)`

`2^(19)-(1)/(2)""^(20)C_(10)`

The sum of the series sum_(r=0) ^(n) "r."^(2n)C_(r), is

`2^(2n -1) `

`n . 2^(2n-1)`

`n 2 ^(n-1)`

`n 2^(2n - 2)`

The sum of the series sum_(r=0) ^(n) ""^(2n)C_(r), is

`2^(2n)`

`2^(n)`

`2^(2n) + ""^(2n)C_(n)`

`(1)/(2) (2^(2n) + ""^(2n)C_(n))`