Question Number 10488 by Saham last updated on 13/Feb/17

An exponential sequence of positive terms and a  linear sequence have the same first term. the sum  o their first term is 3, the sum of their second term  is (3/2), and the sum of their third term is 6. find the  sum of their fifth term.

Answered by mrW1 last updated on 14/Feb/17

terms of A.P.:  a_1 =a  a_2 =a+d  a_3 =a+2d  a_4 =a+3d  a_5 =a+4d  ∙∙∙  terns of G.P.:  b_1 =a  b_2 =a×q  b_3 =a×q^2   b_4 =a×q^3   b_5 =a×q^4   ∙∙∙  a_1 +b_1 =a+a=3  ⇒a=(3/2)  a_2 +b_2 =a+d+a×q=(3/2)  ⇒a+d+aq=(3/2)  ⇒d+(3/2)q=0     ...(i)  a_3 +b_3 =a+2d+a×q^2 =6  ⇒d+(3/4)q^2 =(9/4)     ...(ii)  (ii)−(i):  (3/4)q^2 −(3/2)q−(9/4)=0  q^2 −2q−3=0  (q−3)(q+1)=0  ⇒q=3, since terms of G.P. are positive  d=−(3/2)×3=−(9/2)  a_5 =a+4d=(3/2)+4×(−(9/2))=−((33)/2)  b_5 =a×q^4 =(3/2)×3^4 =((243)/2)  a_5 +b_5 =−((33)/2)+((243)/2)=((210)/2)=105

Commented bySaham last updated on 14/Feb/17

God bless you sir.