InterviewSolution
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InterviewSolution
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प्राकृत संख्या a का मान ज्ञात कीजिए | यदि `Sigma_(k=1)^(n) f(a+k)=16(2^(n)-1)` जहाँ `" " f(x+y)=f(x).f(y).x,y in N` तथा f(1)=2 |
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Answer» दिया है : `" "f(x+y)=f(x).f(y)`व f(1)=2 `:. " " f(2)=f(1+1)=f(1).f(1)=2.2=2^(2)` `f(3)=f(2+1)=f(2).f(1)=2^(2).2=2^(3)` इस प्रकार `"" f(4)=2^(4)`....आदि `:. " " underset(k=1)overset(n)Sigma f(a+k)=underset(k=1)overset(n)Sigma f(a).f(k)=2^(a) underset(k=1)overset(n)Sigma f(k)=2^(a)(2+2^(2)+2^(3)+...2^(n))` `=2^(a+1)[1+2+2^(2)+....n पद]` ` =2^(a+1).(1.(2^(n)-1))/(2-1)` `16(2^(n)-1)=2^(a+1)(2^(n)-1)` `:. " "2^(a+1)=16=2^(4)` `rArr " "a+1=4` `rArr " "a=3` |
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