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.
On equating (1) and (2), we get
0r . This implies
so = = 0.
(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 , then the ratio of term is :
(a) (b) (c) (d)
Sol. (i) (proper method )
Let and be the first term and common difference of first A.P. and and be the first term and common difference of second A.P. , then
Now put . This implies .
(ii) By using formula:
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) (b) 1 (c) (d) none
sol. (i) (proper method ) Let be sum of the ‘n’ terms of the given series , then
Taking limit we get = 3
(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 be positive real numbers such that and , then is equal to (a) (b) (c) (d)
Sol. (proper method) Given that which gives
By applying componendo and dividendo =
By applying componendo and dividend again, we get
= . So the option (a) is correct .
(ii) (by using formula ) Given that .
so and . Now by using formula
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