The first term of a G.P is 2times it's common ratio. Find the sum of the first two terms of a G.P if it's sum to infinity is 8
Let the first term be $2x$ and the common ratio be $x$. Then, since the sum to infinity is $8$, we have:
$$\frac{2x}{1-x}=8$$
Solving for $x$, we get $x=-\frac{2}{3}$ (we reject the positive solution since a common ratio must be less than 1 in absolute value). Thus, the first two terms of the G.P are $2x$ and $2x^2$, which are $-\frac{4}{3}$ and $\frac{8}{9}$ respectively.
Their sum is $-\frac{4}{3}+\frac{8}{9}=-\frac{4}{9}$. Therefore, the sum of the first two terms of the G.P is $\boxed{-\frac{4}{9}}$.
Let's break down the problem step-by-step:
Step 1: Define the terms of the geometric progression (G.P).
Let the first term be "a" and the common ratio be "r".
Given:
First term, a = 2r
Step 2: Find the sum of the first two terms of the G.P.
The sum of the first two terms of a G.P is given by the formula:
Sum of first two terms = a + ar
Substituting the value of a from the given information:
Sum of first two terms = 2r + (2r)(r) = 2r + 2r²
Step 3: Find the infinite sum of the G.P.
The sum of an infinite geometric progression is given by the formula:
Sum to infinity = a / (1 - r)
Substituting the given value of Sum to infinity:
8 = (2r) / (1 - r)
Step 4: Solve the equation.
Divide both sides of the equation by 2r:
4 = 1 / (1 - r)
Take the reciprocal of both sides:
1/4 = 1 - r
Rearrange the equation:
r = 1 - 1/4 = 3/4
Step 5: Find the sum of the first two terms.
Substituting the value of r into the equation:
Sum of first two terms = 2r + 2r² = 2(3/4) + 2(3/4)²
Sum of first two terms = 6/4 + 18/16 = 24/16 + 9/16
Sum of first two terms = 33/16
Answer: The sum of the first two terms of the geometric progression is 33/16.