# Progression practice test

1.

Find  the  15th  term  of  an  arithmetic  progression  whose  first  term  is  2  and  the  common  difference  is  3.

Explanation:

n th  term  of  A.P  =  a +(n-1) *d
=  2+(15-1)*3 ,   =  2 + 42 = 44.

2.

What  is  the  sum  of  the  first  15  terms  of  an  A.P  whose  11 th  and   7 th  terms  are  5.25  and  3.25  respectively

Explanation:

a +10d  = 5.25, a+6d  = 3.25,  4d  =  2,  d  =  ½
a +5  =  5.25, a  = 0.25  = ¼,   s 15 =  15/2 ( 2 * ¼ +  14 * ½ )
=  15/2 (1/2 +14/2 )     =  15/2 *15/2  =225/ 4   =   56.25

3.

If(12+22+32+…..+102)=385,then the value of (22+42+62+…+202) is :

Explanation:

22+42+62+…202=(1*2)2+(2*2)2+(2*3)2+…..+(2*10)2
=22*12+22*22+22*32+…22*102
=22[12+22+….102]
=4*385=1540

4.

In an arithmetic series consisting of 51 terms, the sum of the first three terms is 65 and the sum of the middle three terms is 129. What is the first term and the common difference of the series?

Explanation:

Given a + (a + d) + (a + 2d)= 65
=> 3a + 3d =65…. (1)
The middle terms are 25th , 26th and 27th terms
=> a + 24d + a + 25d + a + 26d = 129
=> 3a + 75d
129 ….(2)
From (1) and (2) d= 8/9    =>a= 187/9

5.

The sum of the first 100 numbers, 1 to 100 is divisible by

Explanation:

The sum of the first 100 natural numbers is given by

= [n * (n + 1) ]/ 2

= [100 * 101]/2

= 50 * 101
101 is an odd number and 50 are divisible by 2.

Hence, 50*101 will be divisible by 2.

6.

How many terms are there in G.P 3,6,12,24,….,384?

Explanation:

Here a=3 and r=6/3=2.Let the number of terms be n
Then,tn=384=>arn-1=384
=>3*2n-1=384=>2n-1=128=27
=>n-1=7=>n=8
There for number of terms=8

7.

Four  angles  of  a  quadrilateral  are  in  G.P.  Whose  common  ratio  is  an  intiger.  Two  of  the  angles  are  acute  while  the  other  two  are  obtuse.   The  measure  of  the  smallest   angle  of  the  quadrilateral  is

Explanation:

Let   the  angles  be  a, ar, ar 2, ar 3.
Sum  of  the angles = a ( r 4- 1 ) /r -1 = a ( r 2 + 1 ) ( r + 1 ) = 360
a< 90 , and  ar< 90,  Therefore,  a ( 1 + r ) <  180,  or   ( r 2 + 1 ) > 2
Therefore, r  is  not  equal  to  1.  Trying  for  r  =  2  we  get  a  = 24  Therefore, The  angles  are  24, 48, 96  and  192.

8.

How many numbers between 11 and 90 divisible by 7?

Explanation:

The required numbers are 14,21,28….84
This is an A.P with a=14 and d=(21-14)=7
Let it contain n terms
Then,Tn=84=>a+(n-1)d=84
=>14+(n-1)*7=84 or n=11
there for required number of terms=11

9.

If Sn denotes the sum of the first n terms in an Arithmetic Progression and S1: S4 = 1: 10

Then the ratio of first term to fourth term is:

Explanation:

Use Sn = (n/2)[ 2a + (n-1)d]   and Tn = a +  (n – 1) d

S1/S4 = 1/10 = a/ (4/2) [2a + 3d]

6a = 6d or a = d

Therefore T1/T4 = a/ (a+ 3d) = a/4a = ¼

10.

The sum of the three numbers in A.P is 21 and the product of their extremes is 45. Find the numbers.