**Arithmetico-geometric Progression(A.G.P.)**

**Definition:**

·The combination of arithmetic and geometric progression is called arithmetico-geometric progression.

*n*^{th}** term of A.G.P.**

· If a_{1},a_{2}, a_{3}, a_{4},.....an is an A.P.

b_{1}, b_{2}, b_{3}....b_{n} is a G.P.,

then the sequence a_{1}b_{1}, a_{2}b_{2}, a_{3}b_{3},.......anbn is said to be an arithmetico-geometric sequence.

Thus, the general form of an arithmetico geometric sequence is ;

a, (a+d)r, (a+2d)r^{2}, (a+3d)r^{3}.................a + (n-1)dr^{(n-1)}

**Sum of A.G.P.**

**(1)**** Sum of n terms:**

The sum of *n* terms of an arithmetico-geometric sequence a, (a+d)r, (a+2d)r^{2}, (a+3d)r^{3}.................a + (n-1)dr^{(n-1)}

**Sum of infinite sequence:**

Let |*r*|< 1. Then r^{n},r^{n-1} as *n* ®¥ and it can also be shown that n r^{n} ®0 as *n* ®¥. So, we obtain that

In other words, when |*r*|< 1 the sum to infinity of an arithmetico-geometric series is ;

**Method for Finding Sum**

· This method is applicable for both sum of *n* terms and sum of infinite number of terms.

· First suppose that sum of the series is *S*, then multiply it by common ratio of the G.P. and subtract. In this way, we shall get a G.P., whose sum can be easily obtained.

· If the differences of the successive terms of a series are in A.P. or G.P., we can find *n*^{th} term of the series by the following steps :

**Step I: **Denote the *n*^{th} term by T_{n} and the sum of the series upto *n* terms by S_{n}.

**Step II: **Rewrite the given series with each term shifted by one place to the right.

**Step III: **By subtracting the later series from the former, find** **T_{n.}

**Step IV: **From T_{n}, S_{n }can be found by appropriate summation.

**Example**:

· Consider the series 1+ 3 + 6 + 10 + 15 +…..to *n* terms.

Here differences between the successive terms are 3-1,6 – 3, 10 – 6, 15 – 10, …….*i.e., * 2, 3, 4, 5,…… which are in A.P. This difference could be in G.P. also. Now let us find its sum

S= 1+3+6 +10 ...........T_{n-1} +T_{n}

S= 1+3 +6 +10..................T_{n-1} +T_{n}

Subtracting, we get ;

0 = 1 +2 + 3 + 4..........( T_{n}-T_{n-1})-T_{n}

T_{n }= 1 +2 + 3 + 4..........up to n terms

**Special Series :**

**Properties of Arithmetic, Geometric, Harmonic means between two given Numbers**

· Let *A*, *G* and *H* be arithmetic, geometric and harmonic means of two numbers *a* and *b*.

Then A = (a+b)/2, G =Öab and H = 2ab/ ( a+ b)

These three means possess the following properties:

A ³ G ³ H Note that the equality holds only when *a* = *b*.

A, G, and H forms a geometric progression such that G^{2} = A x H

The equation having *a* and *b* as its roots is = x^{2}-2Ax+ G^{2} = 0 0r x^{2 }-(a+b)x + ab = 0

The roots a, b are given by : A ± ÖA^{2}-G^{2}

If *A*, *G, H* are arithmetic, geometric and harmonic means between three given numbers *a*, *b* and *c*, then the equation having *a*, *b*, *c* as its roots is ;

**x ^{3}- 3Ax^{2} + 3 G^{3} x/H - G^{3}=0**

**Relation between A.P., G.P. and H.P.**

** (1) **If number of terms of any A.P./G.P./H.P. is odd, then A.M./G.M./H.M. of first and last terms is middle term of series.

**(2) **If number of terms of any A.P./G.P./H.P. is even, then A.M./G.M./H.M. of middle two terms is A.M./G.M./H.M. of first and last terms respectively.

**(3) **If *p*^{th}, *q*^{th} and *r*^{th} terms of a G.P. are in G.P. Then *p*, *q*, *r* are in A.P.

**(4) **If *a*, *b*, *c* are in A.P. as well as in G.P. then a=b=c

**(5) **If *a*, *b*, *c* are in A.P., then x^{a}, x^{b}, x^{c} will be in G.P. x ¹± 1.