Question
The second and third terms of a geometric progression are 10 and 8 respectively. Find the sum to infinity.
Ask by Grant Chambers. in Vietnam
Mar 02,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
The sum to infinity is 62.5.
Solution
Let the first term of the geometric progression be a and the common ratio be r.
The second term is ar = 10 and the third term is ar² = 8.
Step 1. Find the ratio r:
Divide the third term by the second term:
r = (ar²) / (ar) = 8 / 10 = 0.8 (or 4/5)
Step 2. Find the first term a:
Since ar = 10,
a = 10 / r = 10 / 0.8 = 12.5
Step 3. Find the sum to infinity (S∞):
The formula for the sum to infinity of a geometric progression is:
S∞ = a / (1 – r)
Plug in the values:
S∞ = 12.5 / (1 – 0.8) = 12.5 / 0.2 = 62.5
Therefore, the sum to infinity of the geometric progression is 62.5.
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Simplify this solution Bonus Knowledge
In a geometric progression (GP), the second term \( ar = 10 \) and the third term \( ar^2 = 8 \). We can find the common ratio \( r \) by dividing the third term by the second term: \[ \frac{ar^2}{ar} = \frac{8}{10} \implies r = \frac{8}{10} = 0.8. \] Now that we have \( r \), we can find the first term \( a \) using the equation for the second term: \[ a(0.8) = 10 \implies a = \frac{10}{0.8} = 12.5. \] The formula for the sum to infinity of a GP is \( S_\infty = \frac{a}{1 - r} \), given that \( |r| < 1 \). Here, \[ S_\infty = \frac{12.5}{1 - 0.8} = \frac{12.5}{0.2} = 62.5. \] So, the sum to infinity of this geometric progression is \( 62.5 \).
