The sum of the first 20 terms of the series 1 + 3/2 + 7/4 + 15/8 + 31/16 + ... is : (1) 39 + 1/2¹⁹ (

Q: Wait, why is the general term $a_n = 2 - \frac{1}{2^{n-1}}$ and not $2 - \frac{1}{2^n}$? Let's check for $n=1$.

That is a fantastic observation! Let's test it. If $n=1$, then $a_1 = 2 - \frac{1}{2^{1-1}} = 2 - \frac{1}{2^0} = 2 - 1 = 1$. This matches our first term perfectly. If you used $2 - \frac{1}{2^n}$, you would get $2 - 1/2 = 1.5$, which is our second term. Always test your general term formula with $n=1$ to ensure it aligns with the series!

Q: Why can't we just treat this as a standard Arithmetic Progression? It looks like the numerators are increasing.

I understand why you might think that! While the numerators $1, 3, 7, 15$ do follow a pattern, the denominators $1, 2, 4, 8$ are also changing. An Arithmetic Progression requires a constant difference between consecutive terms. Here, the difference between $1$ and $1.5$ is $0.5$, but the difference between $1.5$ and $1.75$ is $0.25$. Since the difference is not constant, it is not an AP. It is a hybrid series that requires us to decompose it into simpler, known structures.

Q: I got $40 - 2 - \frac{1}{2^{19}}$. Why is the final answer $38 + \frac{1}{2^{19}}$?

Ah, the classic sign trap! When we calculate the sum of the GP, we get $2 - \frac{1}{2^{19}}$. Our total sum is $40 - (2 - \frac{1}{2^{19}})$. When you distribute that negative sign, it becomes $40 - 2 + \frac{1}{2^{19}}$. So, $40 - 2$ is $38$, and the negative times negative becomes positive. Keep a close eye on those brackets!

Question

The sum of the first 20 terms of the series $1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots$ is :

Options:

(A) $39 + \frac{1}{2^{19}}$

(B) $39 + \frac{1}{2^{20}}$

(C) $38 + \frac{1}{2^{20}}$

(D) $38 + \frac{1}{2^{19}}$

Sigma Percentile · Accepted Answer

### Analyzing the Setup

When you first look at the series $1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \dots$, it is easy to feel overwhelmed. It is neither a simple Arithmetic Progression nor a standard Geometric Progression.

However, mathematics is rarely about memorizing formulas; it is about finding the hidden order. Let us observe the terms: $1, 1.5, 1.75, 1.875$.

They are all dancing around the number $2$. Specifically, they are all just a tiny bit less than $2$.

### The Art of Observation

Let us rewrite the terms to reveal the underlying structure:
$1 = 2 - 1$
$\frac{3}{2} = 2 - \frac{1}{2}$
$\frac{7}{4} = 2 - \frac{1}{4}$
$\frac{15}{8} = 2 - \frac{1}{8}$

Suddenly, the chaos vanishes. We have found the pattern: every term is simply $2$ minus a fraction.

### The General Term

Now, let us formalize this. We want the $n$-th term, $a_n$. The constant part is always $2$, and the subtracted part involves powers of $2$ in the denominator.

For $n=1$, we subtract $\frac{1}{2^0}$. For $n=2$, we subtract $\frac{1}{2^1}$. For $n=3$, we subtract $\frac{1}{2^2}$.

The exponent is always $n-1$. Thus, our general term is:
$$a_n = 2 - \frac{1}{2^{n-1}}$$

### The Power of Linearity

We need the sum of the first $20$ terms, $S_{20} = \sum_{n=1}^{20} (2 - \frac{1}{2^{n-1}})$. The beauty of the summation operator is its linearity.

We can split this into two separate, manageable tasks:
$$S_{20} = \sum_{n=1}^{20} 2 - \sum_{n=1}^{20} \frac{1}{2^{n-1}}$$

The first part is trivial: adding $2$ to itself $20$ times gives us $40$. The second part is where the real work happens.

### The Geometric Progression

Look at the second summation: $\sum_{n=1}^{20} (\frac{1}{2})^{n-1}$. This is a classic Geometric Progression (GP) where the first term $a=1$ and the common ratio $r=\frac{1}{2}$.

The sum of a GP is given by $S_n = \frac{a(1-r^n)}{1-r}$. Substituting our values, we get:
$$S = \frac{1(1-(\frac{1}{2})^{20})}{1-\frac{1}{2}}$$

This simplifies to $\frac{1-(\frac{1}{2})^{20}}{\frac{1}{2}}$, which is $2(1 - \frac{1}{2^{20}})$. Expanding this, we get $2 - \frac{2}{2^{20}}$, which simplifies beautifully to $2 - \frac{1}{2^{19}}$.

### The Final Synthesis

Now, we bring it all home. Our total sum is $40 - (2 - \frac{1}{2^{19}})$.

Distributing the negative sign, we get $40 - 2 + \frac{1}{2^{19}}$. This results in the final answer:
$$\mathbf{38 + \frac{1}{2^{19}}}$$

There it is! The complexity has been tamed. Remember, whenever you face a series that does not fit a standard mold, look for the relationship between the terms and a nearby constant.

Animated Solution for Mathematics - Sequence and Series: The sum of the first 20 terms of the series $1 + \frac{3}{2} + \frac{7}{4} + \frac{15}{8} + \frac{31}{16} + \dots$ is :

Select Answer:

Visualized Solution

Observe the Series Pattern

Analyze Individual Terms

Find the General Term $a_{n}$

Set up the Summation

Split the Summation

Calculate the Constant Sum

Identify the Geometric Progression

Apply GP Sum Formula

Simplify the GP Sum

Final Calculation

Conclusion and Key Takeaway

The Sigma Insight: Sum of Special Series

Analyzing the Setup

The Art of Observation

The General Term

The Power of Linearity

The Geometric Progression

The Final Synthesis

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