Find the first term of an a.p. if the sum of the first 10 terms is 150 and d =3.

We'll calculate the sum of the first 10 terms of an
arithmetic series using the formula:

S10 = (a1+a10)*10/2,
where a1 is the first term and a10 is the 10th term of the
sum.

We'll write the 10th term with respect to a1 and the
common difference d.

a10 = a1 +
(10-1)*d

a10 = a1 + 9*3

a10 =
a1 + 27

We'll substitute a10 and the value of S10 into the
formula of the sum:

150 = (a1 + a1 +
27)*5

150 = (2a1+27)*5

We'll
remove the brackets and we'll get:

150 = 10a1 +
135

We'll subtract both sides
135:

10a1 = 150 - 135

10a1 =
15

We'll divide by
10:

a1 =
1.5

The first term of the
arithemtic progression, whose sum is 150 and common difference is d = 3, is a1 =
1.5.