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# Confused! kindly explain, - Sequence and series - JEE Main

The sum of the first 20 terms of the series

$1+ \frac{3}{2}+\frac{7}{4}+\frac{15}{8}+\frac{31}{16}+.... ,$

is

• Option 1)

$38+\frac{1}{2^{19}}$

• Option 2)

$38+\frac{1}{2^{20}}$

• Option 3)

$39+\frac{1}{2^{20}}$

• Option 4)

$39+\frac{1}{2^{19}}$

118 Views
N

As we have learned

Sum of n terms of a GP -

$S_{n}= \left\{\begin{matrix} a\frac{\left ( r^{n}-1 \right )}{r-1}, &if \: r\neq 1 \\ n\, a, & if \, r= 1 \end{matrix}\right.$

- wherein

$a\rightarrow$ first term

$r\rightarrow$ common ratio

$n\rightarrow$ number of terms

$S_{20}= 1 +\left ( \frac{4-1}{2} \right )+\left ( \frac{8-1}{4} \right )+\left ( \frac{16-1}{8} \right )+......$

$=1+2+2.......20times....\left ( 1/2+1/4+1/8+......20times \right )$

$39-1/2(\left ( \frac{1-(1/2)^{19}}{1-1/2} \right ))= 39-1+1/(2)^{19}$

$38+\frac{1}{2^{19}}$

Option 1)

$38+\frac{1}{2^{19}}$

This is correct

Option 2)

$38+\frac{1}{2^{20}}$

This is incorrect

Option 3)

$39+\frac{1}{2^{20}}$

This is incorrect

Option 4)

$39+\frac{1}{2^{19}}$

This is incorrect

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