Problema Solution
An exponential sequence(GP) of positive terms and a linear sequence (AP) have the same first term. The sum of their fist terms is 3 the sum of their 2nd terms is 3/2, and the sum of theis 3rd terms is 6. Find the first term, common ratio and common difference.
Answer provided by our tutors
let a,ar,ar2,... be a GP and a,a+d,a+2d,.. be an AP.
a+a=3 ==> a =3/2
ar + a + d = 3/2
==> (3/2)r + 3/2 + d = 3/2
==> (3/2)r = -d
and ar2 + a + 2d = 6
==> (3/2)r2 + 3/2 + 2d = 6
==> 3r2 + 4d = 9
==> 3r2 -6r - 9 = 0
==> (3r-9)(r+1) = 0
==> r = 3 or r = -1
r can not be -1 because GP is of positive terms.
HEnce, r = 3 and d = -(3/2)*3 = -9/2
Hence, GP is 3/2, 9/2, 27/2,.. and AP is 3/2, -3, -15/2..
Hence, 1st term is 3/2, common ratio is 3 and common difference is -9/2