Given the sum of the first 5 terms of an AP as 90 and the first 50 terms as 4275, what is the first term of the series?

We'll apply the formula for the sum of n terms of an
A.P.:

Sn = (a1+an)*n/2

We know
that the sum of the first 5 terms is 90.

S5 =
90

90 = (a1+a5)*5/2

2*90 =
(a1+a5)*5

We'll divide by 5 both
sideS:

2*18 = a1 + a5

We'll
write the formula for the general term of an A.P.

an = a1 +
(n-1)*d

The common difference is
d.

a5 = a1 + (5-1)d

a1 + a5 =
36 (1)

a5 = a1 + 4d (2)

We'll
substitute (2) in (1):

2a1 + 4d =
36

We'll divide by 2:

a1 + 2d
= 18 (3)

We also know that S50 =
4275.

S50 = (a1+a50)*50/2

4275
= (a1+a50)*25

We'll divide by 25 both
sides:

a1 + a50 = 171 (4)

a50
= a1 + 49d (5)

We'll substitute (5) in
(4):

2a1 + 49d  =171 (6)

We'll
form the system from the equation (3) and (6):

a1 + 2d = 18
(3)

2a1 + 49d  =171 (6)

We'll
multiply (3) by -2:

-2a1 - 4d = -36
(7)

We'll add (7) to (6):

-2a1
- 4d + 2a1 + 49d = -36 + 171

We'll eliminate and combine
like terms:

45d = 135

We'll
divide by 45:

d =
3

The common difference is
3.

We'll substitute d in
(3):

a1 + 2d = 18

a1 + 6 =
18

We'll subtract 6 both
sides:

a1 =
12

The first term of the A.P.,
whose common difference is 3 and sum of 5 first terms is 90, is a1 =
12.