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If(1+ x+ x ^ 2+ x^ 3 )^5= sum _(k = 0) ^(15)a _k x ^k, thensum _ (k= 0 ) ^(7) a _(2k)isequalto |
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Answer» 128 ` (1 + x + x ^ 2+ x ^ 3 ) ^5 =sum _(k = 0) ^(15 )=a _ k x ^k ` `RARR (1 +x+x ^ 2+x ^3) ^5 =a_ 0+ a _ 1 x ^ 1+ a _2x ^ 2+… +a _ (14)x ^(14 )+a _ (15)x ^(15)`… (1) Substituting`x = 1`,in (1) ` rArr(1 +1+ 1+ 1 ) ^(5)= a_ 0+ a _ 1 + a _ 2+ ... +a _ (14) +a _ (15)` ` rArr4 ^(5)=a _ 0+a _ 1+a _ 2+...+a _(14)+a _ (15) ""... (2) ` Substituting` x =-1`in (1) ` rArr(1 - 1 + (1) ^2+ (1)^3 =a _0- a _1+ a _2-a_3 +... +a _ (14)-a _ (15) ` `0 =a _ 0- a _ 1+a _ 2 +..+ a _ (14 )+a _ (15) `...(3) Adding(2)and(3) `rArr 4 ^ 5= 2 ( a _ 0+a _ 2+a_4+... +a _ (14)) ` `RARRA _ 0+a _ 2 +... +a _ (14 )= ( 4 ^5 )/(2 ) ` `rArra_ 0+a _ 2+... +a _ (14)=2 ^9 ` ` = 512 ` |
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