The third term of a G.P is 24 and the sum of the first two terms is 288. If all the terms are positive,find
a)the first term and the common ratio
b)the sum of all the terms from 4th term to the 8th term
ar^2 = 24
a + ar = 288
so, we have
r^2 = 24/a
a + a√(24/a) = 288
a = 216
r = 1/3
(b) you want S8-S3, so just plug in your formula for
Sn = a(1-r^n)/(1-r)
To solve this problem, we can use the formula for the nth term of a geometric progression (G.P):
TN = a * r^(n-1)
where TN is the nth term, a is the first term, r is the common ratio, and n is the position of the term in the sequence.
a) Find the first term and the common ratio:
We are given that the third term (T3) is 24:
T3 = a * r^(3-1) = 24
We are also given that the sum of the first two terms is 288:
T1 + T2 = a + a * r = 288
We have two equations with two variables, a and r. We can solve these equations simultaneously:
From the equation T3 = 24, we get: a * r^2 = 24 (Equation 1)
From the equation T1 + T2 = 288, we get: a + a * r = 288 (Equation 2)
We can solve Equation 1 for a by substituting r^2 = 24/a:
a * (24/a) = 24
24 = 24
So, a = 24.
Now we can substitute a = 24 in Equation 2:
24 + 24 * r = 288
24 * (1 + r) = 288
1 + r = 288/24
1 + r = 12
Subtracting 1 from both sides, we get:
r = 12 - 1
r = 11
Therefore, the first term (a) is 24 and the common ratio (r) is 11.
b) Find the sum of all the terms from the 4th to the 8th term:
We can use the formula for the sum of a geometric progression (S_n):
Sn = a * (1 - r^n) / (1 - r)
To find the sum of the terms from the 4th (T4) to the 8th (T8), we need to calculate S8 and S3, then subtract S3 from S8 to get the sum of T4 to T8.
Using the formula, we can find S3:
S3 = a * (1 - r^3) / (1 - r)
S3 = 24 * (1 - 11^3) / (1 - 11)
S3 = 24 * (-1310) / (-10)
S3 = 24 * 131
S3 = 3144
Now let's find S8:
S8 = a * (1 - r^8) / (1 - r)
S8 = 24 * (1 - 11^8) / (1 - 11)
Calculating this expression, we get S8 = -32879985240.
Finally, to find the sum of T4 to T8, we subtract S3 from S8:
Sum of T4 to T8 = S8 - S3 = -32879985240 - 3144 = -32879988384.
Therefore, the sum of all the terms from the 4th term to the 8th term is -32,879,988,384.