The sum of the first 10 terms of a geometric sequence is equal to 1456/243. (a) Find the first term, u₁, if r = %. (b) Find the sum to infinity of this sequence.

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**Geometric Sequences**

The sum of the first 10 terms of a geometric sequence is equal to \( \frac{1456}{243} \).

(a) **Find the first term, \( u_1 \), if \( r = \frac{1}{3} \).**

(b) **Find the sum to infinity of this sequence.**

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### Explanation:

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (\( r \)). The general formula for the sum of the first \( n \) terms of a geometric sequence is:

\[ S_n = u_1 \frac{1-r^n}{1-r} \]

For the sum to infinity, the formula is:

\[ S_{\infty} = \frac{u_1}{1-r} \]

provided that \( |r| < 1 \).
Transcribed Image Text:**Geometric Sequences** The sum of the first 10 terms of a geometric sequence is equal to \( \frac{1456}{243} \). (a) **Find the first term, \( u_1 \), if \( r = \frac{1}{3} \).** (b) **Find the sum to infinity of this sequence.** --- ### Explanation: A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (\( r \)). The general formula for the sum of the first \( n \) terms of a geometric sequence is: \[ S_n = u_1 \frac{1-r^n}{1-r} \] For the sum to infinity, the formula is: \[ S_{\infty} = \frac{u_1}{1-r} \] provided that \( |r| < 1 \).
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