Arithmetico Geometric Series IIT JEE Study Material

An arithmetico geometric series is obtained by term-by-term multiplication of a GP with the corresponding terms of an AP. The general term of an arithmetico geometric series is given by:

For example, the sequence 1 + 3x + 6 x² + 9 x³ + 12 x⁴ + 15 x⁵ + 18 x⁶ is an arithmetico geometric sequence where the sequence 1 + 3 + 6 + 9 + 12 + 15 + 18 is in arithmetic progression and the sequence 1 + x + x² + x³ + x⁴ + x⁵ + x⁶ is in geometric progression.

The Sum of n Terms of an Arithmetico Geometric Series

Now, multiplying the above equation by ‘r’, we get

Now, Equation (2) – Equation (1), we get

Or

Or,

The Sum of an Infinite Arithmetico Geometric Series

If n → ∞, and |r| < 1, then rⁿ = 0.

Therefore,

The Method of Differences:

Suppose p₁, p₂, p₃, p₄,…., p_n is a given sequence such that (p₂ – p₁), (p₃ – p₂),…., (p_n – p_n-1) is either in an arithmetic or geometric progression. The sum of the given sequence can be evaluated by the steps mentioned below:

1. Finding the nth term of the given sequence (t_n):

Let,

Also,

Now, Equation (1) – Equation (2), we get;

Or,

2. Evaluating the sum of the given sequence using its nth term:

Important Results

1. $\begin{array}{l}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}\mathbf{(}{\mathbf{a}}_{\mathbf{r}}\pm {\mathbf{b}}_{\mathbf{r}}\mathbf{)}\mathbf{=}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}{\mathbf{a}}_{\mathbf{r}}\pm \sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}{\mathbf{b}}^{\mathbf{r}}\end{array}$
2. $\begin{array}{l}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}\mathbf{k}{\mathbf{a}}_{\mathbf{r}}\mathbf{=}\mathbf{k}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}{\mathbf{a}}_{\mathbf{r}}\end{array}$
3. $\begin{array}{l}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}\mathbf{k}\mathbf{+}\mathbf{k}\mathbf{+}\mathbf{k}\mathbf{+}\mathbf{k}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\mathbf{n}\mathbf{times}\mathbf{=}\mathbf{n}\mathbf{k}\end{array}$
   [ where, k = constant ]
4. $\begin{array}{l}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}\mathbf{r}\mathbf{=}\mathbf{1}\mathbf{+}\mathbf{2}\mathbf{+}\mathbf{3}\mathbf{+}\mathbf{4}\mathbf{+}\mathbf{5}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}\mathbf{n}\mathbf{=}\frac{\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}}{\mathbf{2}}\end{array}$
5. $\begin{array}{l}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}{\mathbf{r}}^{\mathbf{2}}\mathbf{=}{\mathbf{1}}^{\mathbf{2}}\mathbf{+}{\mathbf{2}}^{\mathbf{2}}\mathbf{+}{\mathbf{3}}^{\mathbf{2}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{n}}^{\mathbf{2}}\mathbf{=}\frac{\mathbf{n}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}\mathbf{(}\mathbf{2}\mathbf{n}\mathbf{+}\mathbf{1}\mathbf{)}}{\mathbf{6}}\end{array}$
6. $\begin{array}{l}\sum _{\mathbf{r}\mathbf{=}\mathbf{1}}^{\mathbf{n}}{\mathbf{r}}^{\mathbf{3}}\mathbf{=}{\mathbf{1}}^{\mathbf{3}}\mathbf{+}{\mathbf{2}}^{\mathbf{3}}\mathbf{+}{\mathbf{3}}^{\mathbf{3}}\mathbf{+}\mathbf{.}\mathbf{.}\mathbf{.}\mathbf{+}{\mathbf{n}}^{\mathbf{3}}\mathbf{=}\frac{{\mathbf{n}}^{\mathbf{2}}\mathbf{(}\mathbf{n}\mathbf{+}\mathbf{1}{\mathbf{)}}^{\mathbf{2}}}{\mathbf{4}}\end{array}$

Video Lessons

Sequence and Series

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Sequence and Series – Revision

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