Question 303 of 347MediumMCQ
Let Sn = 1 + q + q2 + . . . . . + qn and Tn = 1 + \[\left( {{{q + 1} \over 2}} \right) + {\left( {{{q + 1} \over 2}} \right)^2}\] + . . . . . .+ \[{\left( {{{q + 1} \over 2}} \right)^n}\] where q is a real number and q \[ \ne \] 1. If 101C1 + 101C2 . S1 + .... + 101C101 . S100 = \[\alpha \]T100 then \[\alpha \] is equal to
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Official solution
101C1 + 101C2S1 + . . . . . . . + 101C101S100
= \[\alpha \]T100
101C1 + 101C2(1 + q) + 101C3(1 + q + q2) +
. . . . . .+101C101(1 + q + . . . . . + q100)
\[ = 2\alpha {{\left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)} \over {\left( {1 - q} \right)}}\]
\[ \Rightarrow \] 101C1(1 - q) + 101C2(1 - q2) +
. . . . . . + 101C101(1 - q101)
\[ = 2\alpha \left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)\]
\[ \Rightarrow \] (2101 - 1) - ((1 + q)101 - 1)
\[ = 2\alpha \left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)\]
\[ \Rightarrow \] \[{2^{101}}\left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right) = 2\alpha \left( {1 - {{\left( {{{1 + q} \over 2}} \right)}^{101}}} \right)\]
\[ \Rightarrow \] \[\alpha = {2^{100}}\] Answer the question to unlock