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?

Question

#(a_1 + a_2+.....+a_10) / (g_1g_10)# + #(a_2 + a_3 + ....+ a_9) / (g_2g_9)# #+ ....... + (a_5 + a_6) / (g_5*g_6)#

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Answer 1 · 2017-08-25T05:52:32

#(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#

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#