Write the first five terms of the following sequences and obtain the corresponding series
(i) ` a_(1) = 7 , a_(n) = 2a_(n-1) + 3 "for n" gt 1`
(ii)` a_(1) = 2, a_(n) = (a_(n-1))/(n+1) "for n" ge 2 `

Explanation

(i) Given , ` a_(1) = 7, a_(n) = 2a_(n=1) + 3 "for" n gt 1`
` therefore a_(2) = 2a_(1) + 3 = 2 xx7 + 3 = 17`
` a_(3) 2a_(2) + 3 = 37`
` a_(4) = 2a_(3) + 3 = 77`
` a_(4) + 3 = 157`
The corresponding series is : ` 7 + 17 + 37 + 77 + 157 + ...` .
(ii) Given , `a_(1) = 2, a_(n) = (a_(n-1)) /(n+1) . n ge 2 `
` therefore a_(2) = (a_(1))/(3) = (2)/(3)`
` a_(3) = (a_(2))/(4) = (2)/(3) xx(1)/(4) = (1)/(6)` ,
` a_(4) = (a_(3))/(5) = (1)/(6) xx(1)/(5) = (1)/(30)` ,
`a_(5) = (a_(4))/(6) = (1)/(30) xx (1)/(6) = (1)/(180)`
The corresponding series is ,
` 2 + (2)/(3) + (1)/(6) + (1)/(30) + (1)/(180) + ...` .