Progressions

UPSC CSAT Maths

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^n-1)/(r-1) If r > 1, then

Sn=a({1-r}^n )/(1-r) 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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