Describe an infinite geometric series with a beginning value of 2 that converges to 10. What are the first 4 terms of the series?
2/(1-r) = 10
1-r = 1/5
r = 4/5
Take it from there.
To describe an infinite geometric series, we need to first find the common ratio (r), which is the constant multiplier between consecutive terms. In this case, the beginning value (a₁) is 2 and the limit of the series (L) is 10.
The formula for the terms of an infinite geometric series is given by:
aₙ = a₁ * r^(n-1)
where aₙ represents the nth term, a₁ is the first term, r is the common ratio, and n is the position of the term.
To find the common ratio, we can use the formula for the limit of an infinite geometric series:
L = a₁ / (1 - r)
Plugging in the given values, we have:
10 = 2 / (1 - r)
Solving for r, we can rearrange the equation:
10(1-r) = 2
10 - 10r = 2
-10r = -8
r = 8/10
r = 4/5
Now that we have the common ratio (r = 4/5), we can find the first 4 terms of the series by substituting the values into the formula aₙ = a₁ * r^(n-1):
a₁ = 2
a₂ = 2 * (4/5)^(2-1) = 2 * (4/5) = 8/5
a₃ = 2 * (4/5)^(3-1) = 2 * (4/5)^2 = 32/25
a₄ = 2 * (4/5)^(4-1) = 2 * (4/5)^3 = 128/125
Therefore, the first 4 terms of the infinite geometric series are:
2, 8/5, 32/25, 128/125