Sequences and Series Tricks : Arithmetic and Geometric Sequences

Both arithmetic and geometric sequences are important concept of sequences and series . Questions from AP & GP are asked in various competitions including SSC and Railways. Here, are some important formulas that you can use to solve questions based on arithmetic and geometric sequences quickly, easily and efficiently .

**Formula 1. ** ** If for an AP., sum of p terms is equal to sum of q terms then sum of (p+q) terms is zero. **

Que. 1 If the sum of the first 11 terms of an arithmetic progression equals that of the first 19 terms, then what is the sum of the first 30 terms?

(1) 0 (2) –1

(3) 1 (4) Not unique

Sol. (i) ( proper method)

Sum of 11 terms of an AP equals the sum of 19 terms of the same AP.

……….. (i) and

On equating (1) and (2), we get

so =

(ii)( By using formula )

Given that . So

**Formula 2. If the ratio of sum of ‘‘ terms of two arithmetic progression is **

**then the ratio of their ‘ ‘ term will be
.**

Que. 2 If the ratio of sum of terms of two A.P. is

(a)

Sol. (i) (proper method )

Let

Now put . This implies

So .

Hence

(ii) By using formula:

Given

replace ‘n’ with 2n-1,we get

Now put n=12 we get

**Formula 3: sum of infinite series of type **

**= where
and are in A.P. and
**

Que . 3 The sum of the following series is

(a)

sol. (i) (proper method ) Let

=

Taking limit we get

(ii) ( by using formula)

**Formula 4: If arithmetic mean and geometric mean of two numbers ‘a’ and ‘b’ (a>b) are in ratio m:n , then **

Que.4 Let and

Sol. (proper method) Given that

By applying componendo and dividendo =

By applying componendo and dividend again, we get

(ii) (by using formula ) Given that

so and

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