1.Find an A.P. such that the sum of the first three terms is one half the sum of the next four terms,the first term being 12.

Question

2.In an A.P.,the sum of the first three terms is 18,and the sum of the squares of these terms is 126.Find the terms.

Answer 1

Just use the definitions of an AP and translate the English to Math.

"the sum of the first three terms"
= a + a+d + a+2d
= 3a + 3d

"the sum of the next four terms"
= a+3d + a+4d + a+5d
= 3a + 12d

given:
"the sum of the first three terms is one half the sum of the next four terms"
3a+3d = (1/2)(3a+12d)
times 2
6a+6d = 3a+12d
3a=6d
a = 2d
but a = 12, so d =6

the AP is 12, 18, 24, 30, 36, 42, ...

check:
sum of first 3 terms = 12+18+24 = 54
sum of next 3 terms = 30+36+42 = 108
is the first sum one half of the second sum ?

Try the second question in the same way, show me your steps

Answer 2

765