Arithmetic Progression

Quantities are said to be in arithmetic progression when they increase or decrease by a common difference.
E.g., 2, 4, 6, 8, ……. or 3, 7, 11, 15, ……. 
General arithmetic series can be written as,
a, a+d, a+2d, a+3d, a+4d, …… a+nd
(where a is the first term and d is the common difference.) 
The common difference is found by subtracting any term of the series from the next term. That is, the common difference of an A.P. = {tN – (tN – 1)}.

The nth term of an arithmetic progression is given by: Tn =a + (n–1) d.
Finding the sum of the given number of terms in an arithmetic progression
 Let a denote the first term d, the common difference, and n the total number of terms. Also, let L denote the last term, and S the required sum; then
S= n(a+L)/2
L = a + (n – 1) d
S = n/2 [2a+(n–1) d]
Arithmetic Mean
 Let, a and b be two quantities and A be their arithmetic mean.
A=(a+b)/2
Geometric Progression
 Quantities are said to be in Geometric Progression when they increase or decrease by a constant factor. The constant factor is also called the common ratio and it is found by dividing any term by the term immediately preceding it.
 If we examine the series a, ar, ar2 , ar3 , ar4 ,...
 we notice that in any term the index of r is always less by one than the number of the term in the series.
Sum of terms in GP
 Let a be the first term, r the common ratio, n the number of terms, and Sn be the sum to n terms.
Sn=a(r^n1)/(r1) If r > 1, then
Sn=a({1r}^n )/(1r) If r< 1, then
Geometric Mean
Let a and b be the two quantities; G the geometric mean.
G=√ab
Harmonic Progression
 If a, b, c, d is in A.P. then 1/a, 1/b, 1/c and 1/d are all in H.P.
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Practice Questions:
1. Find the sum of all numbers divisible by 6 in between 100 to 400.
a. 12550
b. 12450
c. 11450
d. 11550
e. 11555
2. Find S10 for the following series: 1, 8, 15, ...
a. 325
b. 426
c. 525
d. 225
3. Find the 25th term of the sequence 50, 45, 40, ...
a. 55
b. 65
c. 70
d. 75
4. The 3rd and 8th term of a GP are ⅓ and 81, respectively. Find the 2nd term.
a. 3
b. 1
c. 1/27
d. 1/9
5. What is the nth term of an AP 9, 13, 17, 21, 25, …?
a. 3n + 5
b. 4n + 5
c. 4n + 6
d. 5n + 4
6. How many terms are there in the A.P. 20,25, 30,130?
a. 22
b. 23
c. 21
d. 24
7. In an infinite geometric progression, each term is equal to 3 times the sum of the terms that follow. If the first term of the series is 4, find the product of first three terms of the series?
a. 1/2
b. 1
c. 1/8
d. data insufficient
8. How many natural numbers between 300 to 500 are multiples of 7?
a. 29
b. 28
c. 27
d. 30
9. Rahul saves ₹100 in January 2014 and increases his saving by ₹50 every month over previous month. What is annual saving for Rahul in the year 2014?
a. ₹4200
b. ₹4500
c. ₹4000
d. ₹4100
10. A rubber ball is dropped from a certain height. After striking the floor, the ball bounces to 3/4th of the height it was dropped from. If the ball is initially dropped from a heigh of 100 meters, what is the total distance travelled by the ball before it comes to rest?
a. 400
b. 600
c. 800
d. 700
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