# 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?

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#(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)#

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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 ).

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When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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