Question

The sum of the first 20 terms of the series 1+(3/2) + (7/4) + (15/8) + (31/16) +.....Is

Answer 1

Answer:

the sum of the series is: $=38.0000019073$

Step-by-step explanation:

the series is given by:

$1+\dfrac{3}{2}+\dfrac{7}{4}+.....+\dfrac{37}{2^{19} }$

it could also be written as:

$=1+(2-\dfrac{1}{2^1})+(2-\dfrac{1}{2^{2}})+(2-\dfrac{1}{2^{3}})+....+(2-\dfrac{1}{2^{19}})$

$=1+2\times19-\dfrac{1}{2}(1+\dfrac{1}{2}+\dfrac{1}{2^{2}}+....+\dfrac{1}{2^{18} })$

we will use the sum of geometric progression for finite terms.

$=1+38-\dfrac{1}{2}(\dfrac{1-\dfrac{1}{2^{19} } }{1-\dfrac{1}{2} } )$

$=39-\frac{1}{2}(\dfrac{2^{19}-1 }{2^{17} } )$

$=38.0000019073$

Hence sum of the series is: 38.0000019073