# Determine the common difference and the first term of an A.P. a3=7, a7=8Determine the common difference and the first term of an A.P. a3=7, a7=8

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We know that:

a3= a1+ 2*r = 7........(1)

a7 = a1+ 6r = 8.........(2)

Now all we have to do is solve the system in order to obtain r and a1 values:

By using thre elimination methos, we will subtract (1) from (2):

=> 4*r = 1

**==> r= 1/4**

Now to calculate a1, we will substitute with r=1/4 in either equations:

==> a1+ 3r = 7

==> a1= 7-3r = 7- 3*1/4 = 7-3/4 = 6 1/4 = 25/6

**==> a1= 25/6**

For an A P, the nth term is given as a+(n-1)d where a is the first term and d is the common difference.

Now a7= a +( 7-1 )d= a+6d =8

Also a3 = a + (3-1)d = a +2d =7

Now we have two equations a +6d =8 and a + 2d =7 to solve for a and d.

a+ 6d -a - 2d = 8-7

=> 4d =1

=> d=1/4

Substituting this in a+2d =7

a+ 2*(1/4) =7

=> a = 7- 1/2

=> a= 6.5

**Therefore the first term is 6.5 and the common difference is 1/4**

a3 =7 and a = 8

In an AP, an = a1+(n-1)d, where a1 is the 1st term an is the nth term and d is the common difference.

So a3 = a1+2d = 7

a7 = a1+6d=8

a4-a3 = 6d-2d = 8-7

4d = 1. Or d= 1/4.

Therefore a3 = a1+3d = 7.

a1 = 7-3d = 7-3*(1/4) = (28-3)/4) = 25/4 = 6.25.

Therefore a = 6.25 and d = 1/4 = 0.25.

Let:

d = common difference of the A.P.

Then:

a3 = a1 + 2d = 7 (given)

And

a7 = a1 + 6d = 8

Therefore:

a7 - a3 = 8 - 7 = 1

Substituting above values of a3 and a7 in above equation:

a1 + 6d - (a1 + 2d) = 1

==> 4d = 1

d = 1/4 = 0.25

Substituting this value of d in equation for a3:

d3 = a1 + 2d = 7

a1 +2*1/4 = 7

a1 + 1/2 = 7

a1 = 7 - 1/2 = 6 1/2 = 6.5

Answer:

First term = 6.5

Common difference = 0.25

To calculate the first term of an arithmetic progression, we'll write the general formula for any term of an arithmetic sequence:

an=a1 + (n-1)d, where a1 is the first term and d is the common difference

We'll apply the formula for a3 and a7:

a3=a1 + (3-1)d

a7=a1 + (7-1)d

We'll substitute and we'll get:

8 = a1 + 2rd (1)

20 = a1 + 6d(2)

(2)-(1) =>

8-20 = a1 + 2d-a1 - 6d

We'll eliminate and combine like terms:

-12 = -4d

We'll divide by 4:

**d = 3**

We'll substitute d in (1):

8 = a1 + 2d

8 = a1 + 2x3

8 = a1 + 6

We'll subtract 6 both sides:

a1= 8-6

**a1=2**