Let S_ n =1+q+q²+ .+q^ n and T _ n =1+ ( q +1 2 )+ ( q +1 2 )²+ + ( q +1 2 )^ n where q is a real number and q 1 . If ¹⁰¹ C ₁+ ¹⁰¹ C ₂ S ₁+ .+ ¹⁰¹ C ₁₀₁ S ₁₀₀= T ₁₀₀, then is equal to :

Step-by-step solution

S_ n = ( 1-q^ n+1 1-q ), T_ n = 1- ( q+1 2 )^ n+1 1- ( q+1 2 ) T₁₀₀= 1- ( q+1 2 )¹⁰¹ 1- ( q+1 2 ) S n= 1 1-q - q^ n+1 1-q , T₁₀₀= 2¹⁰¹-( q +1)¹⁰¹ 2¹⁰⁰(1- q ) Now, 101 1 + 101 2 S₁+ 101 3 S₂+ + 101 101 S₁₀₀= ( 1 1-q ) ( 101 2 + + 101 101 ) aligned & - 1 1-q ( 101 2 q²+ 101 3 q³+ + 101 101 q¹⁰¹ )+101 =& 1 1-q (2¹⁰¹-1-101 )- ( 1 1-q ) ((1+q)¹⁰¹-1 . .- 101 1 q )+101 =& 1 1-q [2¹⁰¹-102-(1+q)¹⁰¹+1+101 q ]+101 =& 1 1-q [2¹⁰¹-101+101 q-(1+q)¹⁰¹ ]+101 =& ( 1 1-q ) [2¹⁰¹-(1+q)¹⁰¹ ]=2¹⁰⁰ T₁₀₀ aligned Hence, by comparison =2¹⁰