If a,a_1, a_2,.....a_10,b are in A.P and a,g_1,g_2,g_3................g_10,b are in G.P and h is the H.M between a and b, then find the value of below given expression?

(a_1 + a_2+.....+a_10) / (g_1*g_10) + (a_2 + a_3 + ....+ a_9) / (g_2*g_9) + ....... + (a_5 + a_6) / (g_5*g_6)

1 Answer
Aug 25, 2017

(a_1 + a_2+.....+a_10) / (g_1*g_10) + (a_2 + a_3 + ....+ a_9) / (g_2*g_9) + ....... + (a_5 + a_6) / (g_5*g_6)=30/h

Explanation:

As h is H.M. between a and b, 1/a,1/h,1/b are in A.P. and 1/h=(a+b)/(2ab)

a,a_1, a_2,.....a_10,b are in A.P., we have

a_1+a_10=a_2+a_9=a_3+a_8=...=a_5+a_6=a+b

and as a,g_1,g_2,g_3................g_10,b are in G.P.,

g_1*g_10=g_2*g_9=g_3*g_8=...=g_5*g_6=ab

(a_1 + a_2+.....+a_10) / (g_1*g_10) + (a_2 + a_3 + ....+ a_9) / (g_2*g_9) + ....... + (a_5 + a_6) / (g_5*g_6)

= (5(a+b))/(ab)+(4(a+b))/(ab)+(3(a+b))/(ab)+(2(a+b))/(ab)+((a+b))/(ab)

=(15(a+b))/(ab)

= 15xx2/h

= 30/h