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Find the value of .^20 C0-(.^20C1)/2+(.^...

Find the value of `.^20 C_0-(.^20C_1)/2+(.^20C_2)/3-(.^20C_3)/4+...`

Text Solution

Verified by Experts

The correct Answer is:
`1/21`
`.^(20)C_(0) - (.^(20)C_(1))/(2) + (.^(20)C_(2))/(3) - (.^(20)C_(3))/(4) + "....."`
`= underset(r=0)overset(20)sum(.^(20)C_(r))/(r+1)(-1)^(r)`
`= underset(r=0)overset(20)sum(.^(21)C_(r+1))/(20+1)(-1)^(r)`
` = -1/21 underset(r=0)overset(20)sum.^(21)C_(r+1)(-1)^(r+1)`
`= - 1/21[-.^(21)C_(1) + .^(21)C_(2)-.^(21)C_(3) + "...."]`
`= - 1/(21)[(.^(21)C_(0) - .^(21)C_(1) + .^(21)C_(2) - .^(21)C_(3) + ".....")-.^(21)C_(0) ]`
`= -(1)/(21) [(1-1)^(21) - 1]`
`= 1/21`
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