Question

The sum Of four consecutive numbers in an AP is 32 and the ratio of the product of the first and the last term to the product of two middle terms is 7:15 .How can I find the number ?

Answer 1

we can take four consecutive numbers of AP as a,(a+d)+(a+2d)+(a+3d)..But when we take like this...but we dont know the value of d[d is the common difference] so we will take the AP as (a - 3d), (a - d), (a + d) and (a + 3d) then when we add all these terms we can cancel the term of d and we can find a like this
a-3d + a - d + a + d + a + 3d = 32
4a = 32
a = 32/4
a = 8
from the ap,the first term is (a - 3d) and last term is (a + 3d)....their product is (a² - 9d²) since (a+b)(a-b)=(a²-b²)
value of a=8 so a²=64, (a² - 9d²) =64-9d²------------eqaution 1
two middle terms means 2nd term and 3rd term (a-b)
and (a+b) their product is (a² - d²)=64-d²--------------equation 2
now divide equation 1 and 2
(a - 3d)(a + 3d)/(a - d)(a + d) = 7/15
64-9d²/64-d² = 7/15