$$ 8 + 4 + 2 + 1 + \ldots + \frac { 1 } { 16 } $$

Question

UpStudy AI · Accepted Answer

The given series is:

$$ 8 + 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} $$

This is a **geometric series** where each term is half of the previous one.

**Key Characteristics:**
- **First term ($$ a $$)**: 8
- **Common ratio ($$ r $$)**: $$ \frac{1}{2} $$
- **Number of terms ($$ n $$)**: 8

**Sum of a Geometric Series Formula:**

$$
S_n = a \frac{1 - r^n}{1 - r}
$$

Plugging in the values:

$$
S_8 = 8 \times \frac{1 - \left(\frac{1}{2}\right)^8}{1 - \frac{1}{2}} = 8 \times \frac{1 - \frac{1}{256}}{\frac{1}{2}} = 8 \times \frac{\frac{255}{256}}{\frac{1}{2}} = 16 \times \frac{255}{256} = \frac{4080}{256} = \frac{255}{16}
$$

**Final Answer:**

$$
\frac{255}{16}
$$

Which can also be expressed as:

$$
15 \frac{15}{16}
$$

**Numerical Value:**

$$
255 \div 16 = 15.9375
$$

**Therefore, the sum of the series is $$ \frac{255}{16} $$ or 15.9375.**
The sum of the series $$ 8 + 4 + 2 + 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} $$ is $$ \frac{255}{16} $$ or 15.9375.

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