The fourth term of a GP is 6 and the seventh term is -48,calculate (a)common ratio (b)first term(c)sum of the first eleven terms
-48/6 = r^3
Now you can figure a, and then
S11 = a(r^11-1)/(r-1)
To find the common ratio, we need to understand the formula for the nth term of a geometric progression (GP). The formula for the nth term of a GP is:
an = a1 * r^(n-1),
where an is the nth term, a1 is the first term, r is the common ratio, and n is the position of the term.
Given that the fourth term (a4) is 6 and the seventh term (a7) is -48, we can set up two equations:
a4 = a1 * r^(4-1) = 6 --(1)
a7 = a1 * r^(7-1) = -48 --(2)
To solve for the common ratio (r), divide equation (2) by equation (1):
a7/a4 = [-48/(6 * r^3)] = -8
Simplifying the equation gives us:
r^3 = -48/6 = -8
Taking the cube root of both sides, we have:
r = -2
So the common ratio is -2.
To find the first term (a1), we can substitute the value of r into either equation (1) or (2). Let's use equation (1):
a4 = a1 * (-2)^(4-1) = 6
Simplifying, we get:
a1 * (-2)^3 = 6
a1 * (-8) = 6
a1 = 6 / (-8)
a1 = -3/4
Therefore, the first term (a1) is -3/4.
Finally, to find the sum of the first eleven terms, we can use the formula for the sum of a geometric series:
Sn = a1 * (1 - r^n) / (1 - r)
Substituting the known values, we have:
S11 = (-3/4) * (1 - (-2)^11) / (1 - (-2))
Simplifying:
S11 = (-3/4) * (1 - 2048) / 3
S11 = (-3/4) * (-2047) / 3
S11 = (3/4) * (2047) / 3
S11 = (3 * 2047) / (4 * 3)
S11 = 6141 / 12
Therefore, the sum of the first eleven terms is 514.25.