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Find the value of sumsum(0leiltjlen) (""...

Find the value of `sumsum_(0leiltjlen) (""^(n)C_(i)+""^(n)C_(j))`.

Text Solution

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`underset(0leiltjlen)(sumsum)(.^(n)C_(i)+.^(n)C_(j))=((underset(i=0)overset(n)sumunderset(j=0)overset(n)sum(.^(n)C_(i)+.^(n)C_(j)))-underset(i=0)overset(n)sum2.^(n)C_(i))/2`
`= ((underset(i=0)overset(n)sum(underset(j=0)overset(n)sum.^(n)C_(i)+underset(j=0)overset(n)sum.^(n)C_(j)))-2xx2^(n))/(2)`
` = ((underset(i=0)overset(n)sum(.^(n)C_(i)underset(j=0)overset(n)sum1+2^(n)))-2^(n+1))/(2)`
`= ((underset(i=0)overset(n)sum(.^(n)C_(i)(n+1)+2^(n)))-2^(n+1))/(2)`
`= ((n+1)underset(i=1)overset(n)sum.^(n)C_(i)+2^(n)underset(i=0)overset(n)sum1-2^(n+1))/(2)`
` = ((n+1)2^(n)+2^(n)(n+1)-2^(n+1))/(2)`
` = (n+1)2^(n) - 2^(n) = n2^(n)`
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