KVPY SA Exam 2014 Question Paper

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Question : 62 of 80

Marks: +1, -0

The value of

∑↖{1947}↙{n=0}\ 1/{2^n+√{2^1947

is equal to

$487/√{2^1945}$
$1946/√{2^1947}$
$1947/√{2^1947}$
$1948/√{2^1947}$

Solution:

∑↖{1947}↙{n=0}\ 1/{2^n+√{2^1947

Total terms = 1948

T_1 = 1/{1+√{2^1947}

T_1948 =1/{{2^1947}+√{2^1947}}

T_1 + T_1948 =1/√{2^1947}

Similarly,

T_2 + T_1947 = √{2^1947} = T_3 + T_1946 =

and so on………
Total

1948/2=974

pairs
∴ Sum =

974/√{2^1947}=974/√{4×2^1945}=487/√{2^1945}

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