Question

The common difference of two different arithmetic progressions are equal.The first term of the

first progression is 3 more than the first term of second progression. If the 7th term of first

progression is 28 and 8th term of second progression is 29, then find the both different

arithmetic progressions.​

Answer 1

Step-by-step explanation:

Given:-

The common difference of two different arithmetic progressions are equal.The first term of thefirst progression is 3 more than the first term of second progression. If the 7th term of first progression is 28 and 8th term of second progression is 29.

To find:-

Find the both different arithmetic progressions.

Solution:-

Let the first term of the first AP = a

Let the common difference of the first AP = d

Let the first term of the second AP = b

Then the common difference of the second AP

= d

Since The common difference of two different arithmetic progressions are equal.

Given that

The first term of the first progression is 3 more than the first term of second progression.

a = b+3 ------------(1)

We know that

The general term or nth term of an AP is

an = a+(n-1)d

Now,

the 7th term of first progression is 28

=> a7 = a+(7-1)d = 28

=> a+6d = 28

=> b+3+6d = 28

=> b+6d = 28-3

=> b+6d = 25---------(2)

8th term of second progression is 29.

=> b+7d = 29--------(3)

On Subtracting (2) from (3)

b+7d = 29

b+6d = 25

(-)

_________

0+d = 4

_________

d = 4

Common difference of both AP's = 4

On Substituting the value of d in (2) then

from (2)

b+6(4)=25

=> b+24 = 25

=> b = 25-24

=> b = 1

First term of the second AP = 1

On Substituting the value of b in (1) then

From (1)

a = 1+3

a = 4

First term of the first AP = 4

We know that

The general form of an AP = a,a+d,a+2d,....

First AP :

we have a = 4 and d = 4

a = 4

a+d = 4+4=8

a+2d = 4+2(4) = 4+8 = 12

First AP : 4,8,12,....

Secon AP :

The general form of an AP = b,b+d,b+2d,....

We have b = 1 and d = 4

b= 1

b+d = 1+4=5

b+2d = 1+2(4) = 1+8 = 9

Second AP : 1,5,9,...

Answer:-

First Arithmetic Progression: 4, 8, 12,....

Second Arithmetic Progression : 1, 5, 9,...

Used formulae:-

• The general term or nth term of an AP is

an = a+(n-1)d

• The general form of an AP = a,a+d,a+2d,....