The sum of first 50 term of the series ` 1 + (3 ) /(2) + (7) /(4) + (15 ) /(8) + (31 ) /(16) + `….is `( p + (1)/(2^q )) ` , then value of ` ( p + q ) ` is ……

Answer

147

Explanation

` 1 + (3)/(2) + (7)/(4) + (15)/(8) + (31)/(16) +………..50` terms
` = ( 2-1)/(2^(0)) + (2^(2) -1)/(2) + (2^(3) -1)/(2^(2)) + (2^(4) -1)/(2^(3)) +……….`
` = (2+ 2+……….50` times) - (`1 +(1)/(2) + (1)/(2^(2)) + ……….50` times)
`= 100- {(1-((1)/(2))^(50))/(1-(1)/(2))} = 100 - (2-(1)/(2^(49))) = 98 + (1)/(2^(49))`
`p = 98 , q = 49 rArr p + q = 147`