Harmonic Progression (H.P.)
nth term of HP
A comparison between AP and GP
Arithmetic  Geometric progression
a + (a + d)r + (a + 2d)r^{2 }+ (a + 3d)r^{3} + ……………. Is the form of Arithmetic geometric progression (A.G.P).
One part of the series is in Arithmetic progression and other part is a Geometric progression.
Arithmetic geometric series can be solved as explained in the example below:
Relation between AM, GM and HM:
For two positive numbers a and b
This mean A, G, H are in G.P.
Verifying for numbers 1, 2
Hence
Toolkit
Ex.1 Find the sum of 1 + 2x + 3x^{2} + 4x^{3} + … ∞
Sol. The given series in an arithmeticgeometric series whose corresponding A.P. and G.P. are 1, 2, 3, 4,…
and 1, x, x^{2}, x^{3}, … respectively. The common ratio of the G.P. is x. Let S_{∞} denote the sum of the given
series.
Ex.2 If the first item of an A.P is 12, and 6th term is 27. What is the sum of first 10 terms?
Sol.
Ex.3 If the fourth & sixth terms of an A.P are 6.5 and 9.5. What is the 9th term of that A.P?
Sol.
Ex.4 What is the arithmetic mean of first 20 terms of an A.P. whose first term is 5 and 4th term is 20?
Sol.
Ex.5 The first term of a G.P is half of its fourth term. What is the 12th term of that G.P, if its sixth term
is 6
Sol.
Ex.6 If the first and fifth terms of a G.P are 2 and 162. What is the sum of these five terms?
Sol.
Ex.7 What is the value of r + 3r^{2} + 5r^{3} +     
Sol.
Ex.8 The first term of a G.P. 2 and common ratio is 3. If the sum of first n terms of this G.P is greater
than 243 then the minimum value of ‘n’ is
Sol.
Ex.9
Sol.
Ex.10
Sol.
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