Question

The sixth term of an appointment is 37 and the sum of the first sixth term is147 find the first term,find the sum of the first fifteen terms.

Answer 1

Surely you mean "arithmetic progression" not "appointment"!

a+5d = 37
6/2 (2a+5d) = 147

You want

15/2 (2a+14d)

so solve for a and d and plug 'em in.

Answer 2

a + 5d = 37 ... Eqn 1
6a + 15d = 147 ... Eqn 2
Solve eqn. 1 and 2 simultaneously
a = 12
d = 5
First term = 12

S(15) = 15/2[2a + 14d]
S(15) = 15/2[(2x12) + (14x5)]
S(15) = 705
sum of the first fifteen terms is 705