Sequences

PERFORM C P T E C O M A T H S T A T COURSE BEST IN TOWN! CALL 9331261452 SEQUENCE COURSE COMPLIMENT Formula

'n' th term Common difference / ratio Sum of 'n' terms

ARITHMATIC PROGRESSION (A.P.)

t n=a n−1∗d

GEOMETRIC PROGRESSION (G.P.)

t n=a∗r  n−1

'd' = next term – previous term

n S n = ∗ al where l=t n 2 n S n = 2∗an−1∗d  2 SPECIAL SUMMATIONS

'r' = next term / previous term

a∗1−r n 1−r  a∗ r n −1 Sn=  r−1 Sn=

when r < 1 when r > 1

SUM OF INFINTE GEOMETRIC SERIES

Sum of 1st n Natural Numbers (1+2+3+4+..........n)

n S n = ∗n1 2

(ONLY WHEN R < 1)

S ∞=

a 1−r 

Sum of 1st n Odd Numbers {1+3+5+7+9+............+(2n - 1)}

S n =n 2 Sum of 1st n Even Numbers (2+4+6+...............+2n)

S n =n∗ n1

Sum of squares of 1st n Natural Numbers 2

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2

2

1 2 3 ............n  n∗n1∗2n1 Sn= 6 Sum of cubes of 1st n Natural Numbers -

132 333...........n3  n∗ n1 2 S n =[ ] 2 Mean

Arithmetic Mean of two numbers 'a' and 'b' =

ab 2

Instructor: Ashani Dasgupta

Geometric Mean of two numbers 'a' and 'b' =

± a∗b To insert 'n' Geometric Means between two

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PERFORM C P T E C O M A T H S T A T COURSE BEST IN TOWN! CALL 9331261452 SEQUENCE COURSE COMPLIMENT To insert 'n' Arithmetic Means between two numbers 'a' and 'b' -

d= Derived Progression

If

numbers 'a' and 'b' -

b  1  r =   n1 (r = common ratio) a

b−a  ( d = common difference)  n−1

a 1, a 2, a 3 are in A.P. then 1. a 1±c , a 2±c , a 3±c 2. a 1∗c , a 2∗c , a3∗c a 1 a 2 a3 3. , , c c c

are also in A.P.

If

a 1, a 2, a 3 are in G.P. Then 1. a 1∗c , a 2∗c , a3∗c a 1 a 2 a3 2. , , c c c 1 1 1 3. a 1, a 2, a 3 4. a m1 , a m2 , a m3 ('m' is non zero real)

are also in G.P. TYPE OF SUM PROBLEMS 1 0.6+0.66+0.666+0.6666.............................. Take

6 common and then subtract each term from 1. 9

2 1+12+123+1234+.................................. S n = 1+12+123+1234+........................

Sn =

1+ 12+ 123+........................ Subtraction will give t n Sum the

t n terms

3 Difference between consecutive terms in G.P. Expand and generalize

4 Arithmetic-Geometric Progression Multiply

S n by common ratio.

Instructor: Ashani Dasgupta

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