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Harmonic Progression (H.P.)

- A series of quantities is said to be in a harmonic progression when their reciprocals are in arithmetic progression.

- e.g. are in HP as their reciprocals

- 3, 5, 7, …, and a, a + d, a + 2d….…. are in AP.

nth term of HP

- Find the nth term of the corresponding AP and then take its reciprocal.

- If the HP be

- Then the corresponding AP is a, a + d, a + 2d, ……

- T
_{n}of the AP is a + (n - 1) d - T
_{nth}of the HP is ……

- In order to solve a question on HP, one should form the corresponding AP.

**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 arithmetic-geometric 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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