Three numbers whose sum is 15 are in a.p. if 1,4,19 be added to them respectively the resulting number are in g.p find n
Let the original numbers be
a, a+d, and a+2d
3a + 3d = 15 or a+d = 5 ----> d = 5-a
after the addition, the three numbers are:
a+1, a+d+4, and a+2d+19
they are now in GP, that is ....
(a+d+4)/(a+1) = (a+2d+19)/(a+d+4)
(a + 5-a + 4)/(a+1) = (a + 10-2a + 19)/(a + 5-a + 4)
9/(a+1) = (-a + 29)/9
81 = -a^2 + 28a + 29
a^2 - 28a + 52 = 0
(a - 26)(a - 2) = 0
a = 26 or a = 2
if a = 26, then d = 5-26 = -21
and the original 3 numbers were:
26, 5, and 16
if a = 2, then d = 5-2 = 3
and the original 3 numbers were:
2, 5, and 8
Your question was "find n", I will let you figure out what that means.
2,5,8, or 26,5,-16
good answer
how did the difference 9 come
Thank🙏💕 you
To find the value of "n" in this problem, we need to break it down step by step. Let's start by understanding the given information.
We are given that three numbers whose sum is 15 are in arithmetic progression (AP). Let's assume the three numbers are a, a+d, and a+2d, where "a" is the first term and "d" is the common difference.
Next, we are told that if 1, 4, and 19 are added to these three numbers respectively, the resulting numbers are in geometric progression (GP).
Let's denote the resulting numbers as (a+1), (a+d+4), and (a+2d+19). We can set up the following equation based on this information:
(a+d+4)/(a+1) = (a+2d+19)/(a+d+4)
To simplify this equation, we can cross-multiply:
(a+d+4)*(a+d+4) = (a+1)*(a+2d+19)
Expanding both sides of the equation:
(a^2 + 2ad + ad + d^2 + 8a + 8d + 16) = (a^2 + ad + 19a + 2ad + 2d + 19)
Combining like terms:
a^2 + 2ad + ad + d^2 + 8a + 8d + 16 = a^2 + ad + 19a + 2ad + 2d + 19
Canceling out the common terms:
d^2 + 8a + 8d + 16 = ad + 19a + 2ad + 2d + 19
Rearranging the equation:
d^2 - ad + 8a - 19a + 2ad + 8d - 2d = 19 - 16
Simplifying further:
d^2 + (8-19+2a-2)a + (8-2+8-2)d = 3
d^2 + (2a-11)a + (12)d = 3
Since the resulting equation involves both "a" and "d," we cannot solve for "n" directly without additional information or constraints. It is not possible to determine the value of "n" based on the given information.