ARITHMETIC AND GEOMETRIC PROGRESSION QUESTION AND ANSWERS

About "Arithmetic and Geometric Progression Question and Answers"

Arithmetic and Geometric Progression Question and Answers :

Here we are going to see some practice questions on arithmetic and geometric progression question and answers.

Question 1 :

If tk is the kth term of a GP, then show that tn−k, tn, tn+k also form a GP for any positive integer k.

Solution :

tn  =  arn-1 

kth term :

tk  =  ark-1

(n-k)th term :

tn-k  =  arn-k-1  -----(1)

nth term :

tn  =  arn-1  -----(2)

(n+k)th term :

tn+k  =  arn+k-1  -----(3)

In order to prove the above terms are in G.P, we have to show that the common difference are same.

(2)/(1)  ==> arn-1/arn-k-1  =  rn-1-n+k+1 =  rk

(3)/(2)  ==> arn+k-1/arn-1  =  rn+k-1-n+1 =  rk

Since the common difference are same, the above terms are in G.P.

Question 2 :

If a, b, c are in geometric progression, and if a1/x = b1/y = c1/z, then prove that x, y, z are in arithmetic progression.

Solution :

a1/x = b1/y = c1/z  =  k

If x, y and z are in A.P, then 2y  =  x + z.

In order to prove the above relationship,

a1/x =  k

a  =  kx

b1/y =  k

b  =  ky

c1/z =  k

c  =  kz

Since a, b and c are in G.P

b/a  =  c/b

b2  =  ac

 (ky)2  =   kx kz

 k2y  =   k(x+z)

2y  =  x + z

Hence x,y and z are in A.P

Question 3 :

The AM of two numbers exceeds their GM by 10 and HM by 16. Find the numbers.

Solution :

Let the required numbers be x and y

AM  =  GM + 10   --(1)

AM  =  HM + 16   --(2)

(1)  =  (2)

GM + 10  =  HM + 16

HM  =  GM + 10 - 16

HM  =  GM - 6 --(3)

(GM)2  =  AM (HM)

By applying the values of AM and HM in terms of G.M, we get

(GM)2  =  (GM + 10) (GM - 6)

GM2  =  GM2 + 4GM - 60

4GM  =  60

GM  =  15

By applying the value o GM in (3), we get HM

HM  =  15 - 6  ==>  9

AM  =  15 + 10  =  25

√ab  =  15

ab  =  225

b  =  225/a

a+b/2  =  25

a + b  =  50

a + (225/a) = 50

a2 + 225  =  50a

a2 - 50 a + 225  =  0

(a - 5)(a - 45)  =  0

a  =  5 and a  =  45

If a  =  5

b =  225/5  =  45

If a  =  45

b =  225/45  =  5

Hence the required numbers are 5 and 45.

After having gone through the stuff given above, we hope that the students would have understood "Arithmetic and Geometric Progression Question and Answers"

Apart from the stuff given above, if you want to know more about "Arithmetic and Geometric Progression Question and Answers". Apart from the stuff given in this section, if you need any other stuff in math, please use our google custom search here.

Kindly mail your feedback to v4formath@gmail.com

We always appreciate your feedback.

©All rights reserved. onlinemath4all.com

Recent Articles

  1. SAT Math Videos

    May 22, 24 06:32 AM

    sattriangle1.png
    SAT Math Videos (Part 1 - No Calculator)

    Read More

  2. Simplifying Algebraic Expressions with Fractional Coefficients

    May 17, 24 08:12 AM

    Simplifying Algebraic Expressions with Fractional Coefficients

    Read More

  3. The Mean Value Theorem Worksheet

    May 14, 24 08:53 AM

    tutoring.png
    The Mean Value Theorem Worksheet

    Read More