Sum of 4 terms in GP is 312 . Sum of 1st and 4th term is 252. Find the numbers.
It's in GP and not in AP
To find the numbers in a geometric progression (GP), we will use the formula for the nth term of a GP:
a_n = a * r^(n-1)
where a is the first term, r is the common ratio, and n is the number of terms.
Let's represent the four terms in the GP as a, ar, ar^2, and ar^3.
Now, we are given two pieces of information:
1. The sum of the four terms is 312:
a + ar + ar^2 + ar^3 = 312
2. The sum of the first and fourth terms is 252:
a + ar^3 = 252
We can use these two equations to solve for the values of a and r.
First, let's simplify the equation for the sum of the four terms:
a(1 + r + r^2 + r^3) = 312
Next, let's substitute the second equation into the first equation:
a(1 + r + r^2 + (252 - a)) = 312
We can now simplify and solve for a:
a(253 + r + r^2) = 312
253a + ar + ar^2 = 312
ar + ar^2 = 312 - 253a
Using the formula for the sum of an infinite geometric series:
S = a / (1 - r)
we can substitute the values of a and r to solve for the common ratio:
252 = a / (1 - r)
252(1 - r) = a
Now, substitute this value of a in the equation ar + ar^2 = 312 - 253a:
r(252 - 252r + r^2) = 312 - 253(252 - 252r + r^2)
252r - 252r^2 + r^3 = 312 - 64056 + 64056r - 64056r^2 + 253r^3
0 = r^3 - 64056r^2 + 64248r - 63744
Using a numerical method or factoring, we can find that r = 3.
Substituting r = 3 into the second equation:
252(1 - r) = a
252(1 - 3) = a
-504 = a
So, the four terms in the GP are:
a = -504
ar = -1512
ar^2 = -4536
ar^3 = -13608
Therefore, the numbers in the GP are -504, -1512, -4536, and -13608.
oops. well, use the GP formulas instead. Same idea.
4/2 (2a+3d) = 312
a + a+3d = 252
now solve for whatever numbers you are interested in.