# Sequences And Series

For better understanding of this chapter,first we have to clear what are sequence, series and progression. Let's know about sequence, series and progression.
Sequence - A sequence is an arrangement of numbers seperated by commas.
1, 2, 3, 4, 5,...........................
2, 4, 6, 8, 10,.........................
5, 10, 101, 204, 506,..................

The different numbers in the sequence are called terms and denoted by T1, T2, T3, ............
or a1, a2, a3, …………… Here Tn or an is called the nth term.
In the above 3 examples first two sequences have a fixed formula to find the next term like in first example  an term is n where n = 1, 2, 3,4, …… etc.
In the second example an term is 2n
i.e. for first term n = 1
2 x 1 = 2
for second term n = 2
2 x 2 = 4 … etc
But in the third sequence 5, 10, 101, 204, 506, …… there is no fixed formula for general term.
that means in the sequences there may be a fixed formula for the nth term or maybe it’s not.

Series - If the terms of the sequence connected by positive or negative sign form a series.
For example - 1 + 2 + 3 + 4 + …………..

Progression - Sequences following definite patterns are called progression.
OR
If the terms of a sequence can be described by a fixed formula, then the sequence is called
a progression.
Types of Sequences -
i) Finite Sequences - Numbers of terms are finite.
Example - 2, 4, 5, 6, 7, 8 (Here no. of terms are six.)

ii) Infinite Sequences - Numbers of terms are infinite.
Example - 3, 6, 9, 12, 15, ……………….. (Here we can not count the terms of this sequence, hence it is an infinite sequence)
.

## INTRODUCTION OF ARITHMETIC PROGRESSION

(AP) An arithmetic progression is the list of the numbers in which each term is obtained by adding a fixed number to the preceeding term except the first term. This fixed number is called the common difference (d) of the AP. It can be positive ,negative or zero.

### General form of an AP :

$a,a+d,a+2d,a+3d,.....$

## SOME IMPORTANT QUESTIONS OF ARITHMETIC PROGRESSIONS

1. Show that the sequence < an > defined by an = 4n + 5 is an A.P. Also, find its common difference.

2. Show that the sequence < an > is an A.P. if its nth term is a linear expression in n and in such a case the common difference is equal to the coefficient of n.

3. Show that the sequence Forms an A.P.

4. Show that the sequence log a, log(ab), log(ab2), log(ab3),..... Is an A.P. Find its nth term.

5. Is 184 a term of the sequence 3, 7, 11, ….?

6. Which term of the sequence 20, 1914, 1812…… is the first negative term?

7. Which term of the sequence 8 - 6i, 7 - 4i, 6 - 2i,.......... Is

(i) purely real                    (ii) puerly imaginary?

8. Show that there is no A.P. which consists only of distinct prime numbers.

9. How many numbers of two digits are divisible by 7?

10. In an A.P. the sum of the terms equidistant from the beginning and the end is always same and equal to the sum of first and last terms?

11. In the A.P. 2, 5, 8,...... up to 50 terms, and 3, 5, 7, 9,........ Upto 60 terms, find how many terms are identical.

12. If a1, a2, a3,....... , an are in A.P., where ai > 0 for all i, show that$\frac{1}{\sqrt{{a}_{1}}+\sqrt{{a}_{2}}}\text{\hspace{0.17em}}+\text{\hspace{0.17em}}\frac{1}{\sqrt{{a}_{2}}+\sqrt{{a}_{3}}}\text{\hspace{0.17em}}+.........+\frac{1}{\sqrt{{a}_{n-1}}+\sqrt{{a}_{n}}}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\frac{n-1}{\sqrt{{a}_{1}}+\sqrt{{a}_{n}}}$

13. The first and the last terms of an A.P. are a and l respectively. Show that the sum of nth term from the beginning and nth  term from the end is a + l.

14. Divide 32 into four parts which are in A.P. such that the product of extremes is to the product of means is 7 : 15.

15. If the sum of three numbers in A.P. is 24 and their product is 440, find the numbers.

16. Find the sum of all odd integers between 2 and 100 divisible by 3.

17. Prove that a sequence is an A.P. iff the sum of its n terms is of the form An2  + Bn, where A, B are constants.

18. If the sum of n terms of an A.P. is 3n2 + 5n and its mth term is 164, find the value of m.

19. The sum of the first four terms of an A.P. is 56. The sum of the last four terms is 112. If its first term is 11, then find the number of terms.

20. If the first term of an A.P. is 2 and the sum of first five terms is equal to one -fourth of the sum of the next five terms, find the sum of first 30 terms.

21. The pth term of an A.P. is a and the qth term is b. Prove that the sum of its ( p +q) terms is $p+q 2 { a+b+ a−b p−q }$
22. The sum of n terms of the two arithmetic progressions are in the ratio
(3n + 8) : ( 7n + 15). Find the ratio of their 12th terms.