SEQUENCES and SERIES formulas

 SEQUENCES and SERIES Formulas

Sequences:- A finite sequence is a sequence that contains the last term such as a1, a2, a3, a4, a5, a6……an. On the other hand, an infinite sequence is never-ending i.e. a1, a2, a3, a4, a5, a6……an…..

AP a1, a2, a3, a4, a5, a6……an…..

GP:- a, ar, ar2, ar3, ar4,…,an

Arithmetic Progression

 In AP, we will come across three main terms, which are denoted as:

• Common difference (d)
• nth Term (an)
• Sum of the first n terms (Sn)

Nth term of sequence is a + (n – 1) d,

an = a + (n − 1) × d

Geometric Progression:-

General Form of GP

The general form of Geometric Progression is:

a, ar, ar2, ar3, ar4,…,an

Where,

a = First term

r = common ratio

an = nth term

Formula :-

an = arn-1

To find the sum of the first $n$ terms of a geometric sequence(G.P) sequence. use the formula,
Sn=a1(1−rn)1−r,  r≠1
where $n$ is the number of terms, a1 is the first term and r is the common ratio .

Series:-In a finite series, a finite number of terms are written like a1 + a2 + a3 + a4 + a5 + a6 + ……an. In case of an infinite series, the number of elements are not finite i.e. a1 + a2 + a3 + a4 + a5 + a6 + ……an +

Sum of n terms in AP	n/2[2a + (n – 1)d]
Sum of natural numbers	n(n+1)/2
Sum of square of ‘n’ natural numbers 12 + 22 + 32 + 42 + ………. + n2	[n(n+1)(2n+1)]/6
Sum of Cube of ‘n’ natural numbers 13 + 23 + 33 + 43 + ………. + n3	[n(n+1)/2]2

SUM no:1- Consider the sequence 1, 4, 16, 64, 256, 1024….. Find the common ratio and 9th term. 

SUM no:2- If 4,7,10,13,16,19,22……is a sequence, Find:

1. Common difference
2. nth term
3. 21st term

sum no: 3_ :

Find the sum of the first 40

4040 terms of the arithmetic sequence2,5,8,11,14,⋯

Sum no: 4:-

Find the sum of the first 8 terms of GP series. A₁

 = 1 and r= 2.

SUM no :-5 At the end of each year the value of a certain machine has depreciated by 20% of its value at the beginning of that year. If its initial value was Rs 1250, find the value at the end of 5 years.

SUM NO:- 6 Find the nth term of AP: 1, 2, 3, 4, 5…., an, if the number of terms are 15.

SUM NO:7:-

Write down the 8th term in the Geometric Progression 1, 3, 9, ...

SUM NO: 8- A manufacturer reckons that the value of a machine, which costs him rs.15625, will depreciate each year by 20%. find the estimated value of at the end of 5 years.

rms of the geometric series if first term is 1 and r=2.

00re$r=2$ .