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)#

Caleb Alexander · Accepted Answer

#(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#

Madelyn Anderson · Answer

undefined To find the value of the given expression, we first need to determine the specific terms in the arithmetic progression (A.P.), geometric progression (G.P.), and the harmonic mean (H.M.) between $ a $ and $ b $.

In an arithmetic progression, each term is obtained by adding a common difference to the preceding term. Let's denote the common difference of the arithmetic progression as $ d $. So, the terms of the arithmetic progression are $ a, a_1 = a + d, a_2 = a + 2d, \ldots, a_9 = a + 9d, b = a + 10d $.

In a geometric progression, each term is obtained by multiplying the preceding term by a common ratio. Let's denote the common ratio of the geometric progression as $ r $. So, the terms of the geometric progression are $ a, g_1 = ar, g_2 = ar^2, \ldots, g_9 = ar^9, b = ar^{10} $.

The harmonic mean (H.M.) between $ a $ and $ b $ is given by $ h = \frac{2ab}{a + b} $.

Now, let's denote the given expression as $ E $. Substituting the values of the arithmetic progression, geometric progression, and harmonic mean, we get:

$$ E = a + a_1 + a_2 + \ldots + a_9 + b + g_1 + g_2 + \ldots + g_9 + b + h $$

$$ = (a + a_1 + a_2 + \ldots + a_9 + b) + (g_1 + g_2 + \ldots + g_9 + b) + h $$

$$ = (10a + 55d) + (a(1 + r + r^2 + \ldots + r^9) + b) + h $$

$$ = (10a + 55d) + (a\frac{r^{10} - 1}{r - 1} + b) + h $$

$$ = (10a + 55d) + (\frac{a(r^{10} - 1)}{r - 1} + ar^{10}) + h $$

$$ = (10a + 55d) + (\frac{a(r^{10} - 1) + ar^{10}(r - 1)}{r - 1}) + h $$

$$ = (10a + 55d) + (\frac{a(r^{10} - 1 + r^{11} - r)}{r - 1}) + h $$

$$ = (10a + 55d) + (\frac{a(r^{10} - r + r^{11} - 1)}{r - 1}) + h $$

$$ = (10a + 55d) + (\frac{a(r^{10} - r(1 - r^{10}) - 1)}{r - 1}) + h $$

$$ = (10a + 55d) + (\frac{a(r - 1)(r^{10} - 1)}{r - 1}) + h $$

$$ = (10a + 55d) + (a(r^{10} - 1)) + h $$

$$ = 10a + 55d + ar^{10} - a + h $$

$$ = (10a + ar^{10}) + 55d - a + h $$

$$ = (10a + ar^{10}) - a + 55d + h $$

$$ = (10a + ar^{10} - a) + 55d + h $$

$$ = (9a + ar^{10}) + 55d + h $$

So, the value of the given expression is $ (9a + ar^{10}) + 55d + h $.