Given that the sum of infinity of a G.P is 81. If the first term of the G.P is 9. Determine the common ratio and third term
9/(1-r) = 81
knowing r, then a3 = ar^2
To find the common ratio (r) and the third term (a3) of a geometric progression (G.P) where the sum of the infinite series is 81, and the first term (a1) is 9, we can use the formula for the sum of an infinite G.P:
Sum = a1 / (1 - r)
Given that the sum is 81 and a1 is 9, we can substitute these values into the formula:
81 = 9 / (1 - r)
To find the common ratio, we can rearrange the equation:
1 - r = 9 / 81
1 - r = 1 / 9
To isolate r, we subtract 1 from both sides:
-r = 1 / 9 - 1
-r = (1 - 9) / 9
-r = -8 / 9
Multiplying both sides by -1:
r = 8 / 9
Therefore, the common ratio of the geometric progression is 8 / 9.
To find the third term (a3), we can use the formula for a term in a G.P:
an = a1 * (r)^(n-1)
Substituting the known values:
a3 = 9 * (8 / 9)^(3-1)
a3 = 9 * (8 / 9)^2
a3 = 9 * (64 / 81)
Simplifying:
a3 = 576 / 81
a3 = 64 / 9
Therefore, the third term of the geometric progression is 64 / 9.
To solve this problem, we need to use the formula for the sum of an infinite geometric progression (G.P).
The formula for the sum of an infinite G.P is S = a / (1 - r), where S is the sum, a is the first term, and r is the common ratio.
Given that the sum of the infinite G.P is 81 and the first term is 9, we can substitute these values into the formula:
81 = 9 / (1 - r)
Now, let's solve for the common ratio, r:
Multiply both sides of the equation by (1 - r):
81(1 - r) = 9
Expand the equation:
81 - 81r = 9
Move the constant term to the other side:
-81r = 9 - 81
Simplify:
-81r = -72
Divide both sides of the equation by -81:
r = -72 / -81
Simplify:
r = 8/9
So, the common ratio (r) is 8/9.
To find the third term of the G.P, we can use the formula aₙ = a * r^(n-1), where aₙ is the nth term, a is the first term, r is the common ratio, and n is the position of the term.
Since we know the first term is 9, and the common ratio is 8/9, we can substitute these values into the formula to find the third term (a₃):
a₃ = 9 * (8/9)^(3-1)
Simplify:
a₃ = 9 * (8/9)^2
Evaluate the exponent:
a₃ = 9 * (64/81)
Multiply the fractions:
a₃ = 576/81
Simplify the fraction:
a₃ = 64/9
So, the third term of the G.P is 64/9.