Question

Sum to n terms is

• A. $\frac { 1 } { 2 }$
• B. $\frac { 1 } { 2 }$
• C. $\frac { 1 } { 2 }$ $\left[ 1 - \frac { 1 } { 1.3 .5 \ldots \ldots \ldots . ( 2 n + 1 ) } \right]$
• D. None of these

Answer 1

Answer: (C)

$\frac { 1 } { 2 }$ $\left[ 1 - \frac { 1 } { 1.3 .5 \ldots \ldots \ldots . ( 2 n + 1 ) } \right]$

Answer 2

t_n = $\frac { \mathrm { n } } { 1.3 .5 \ldots \ldots \ldots . ( 2 \mathrm { n } + 1 ) }$

= $\frac { 1 } { 2 }$.

=$\frac { 1 } { 2 }$ $\left[ \frac { 1 } { 1.3 \cdot 5 \ldots \ldots . ( 2 n - 1 ) } - \frac { 1 } { 1.3 .5 \ldots \ldots . ( 2 n + 1 ) } \right]$

=$\frac { 1 } { 2 }$(T_n–1– T_n)

\ 2t_n = T_n–1 –T_n …(1)

2S_n = = (T₁–T₂) + (T₂ –T₃) + ……+T_n–1 –T_n

Ž 2(S_n – t₁) = T₁ –T_n Ž 2S_n = 2t₁ + T₁ –T_n

Ž 2S_n = 2. $\frac { 1 } { 1.3 }$+ $\frac { 1 } { 1.3 }$ – $\frac { 1 } { 1.3 .5 \ldots \ldots . ( 2 n + 1 ) }$

Ž S_n = $\frac { 1 } { 2 }$ $\left[ 1 - \frac { 1 } { 1.3 .5 \ldots \ldots \ldots . ( 2 n + 1 ) } \right]$