A sequence of number is represented as `a_1, a_2, a_3,….a_n`. Each number in the sequence (except the first and the last) is the mean of the first two adjacent numbers in the sequece. If `a_(1) = 1 and a_5 = 3`, what is hte value of `a_(3)`?

Options

A. `1//2`
B. `1`
C. `3//2`
D. `2`

Answer

To solve the problem, we need to find the value of \( a_3 \) in the sequence defined by the conditions given. Let's break it down step by step.

### Step 1: Understand the sequence
We have a sequence represented as \( a_1, a_2, a_3, a_4, a_5 \). We know:
- \( a_1 = 1 \)
- \( a_5 = 3 \)

### Step 2: Write down the relationships
According to the problem, each number in the sequence (except the first and last) is the mean of its two adjacent numbers. Thus, we can express \( a_2 \) and \( a_4 \) in terms of \( a_3 \):
- \( a_2 = \frac{a_1 + a_3}{2} = \frac{1 + a_3}{2} \)
- \( a_4 = \frac{a_3 + a_5}{2} = \frac{a_3 + 3}{2} \)

### Step 3: Express \( a_3 \) in terms of \( a_2 \) and \( a_4 \)
Since \( a_3 \) is also the mean of \( a_2 \) and \( a_4 \), we have:
\[ a_3 = \frac{a_2 + a_4}{2} \]

### Step 4: Substitute \( a_2 \) and \( a_4 \)
Now we substitute the expressions for \( a_2 \) and \( a_4 \) into the equation for \( a_3 \):
\[ a_3 = \frac{\left(\frac{1 + a_3}{2}\right) + \left(\frac{a_3 + 3}{2}\right)}{2} \]

### Step 5: Simplify the equation
First, combine the terms in the numerator:
\[ a_3 = \frac{\frac{1 + a_3 + a_3 + 3}{2}}{2} \]
\[ = \frac{\frac{4 + 2a_3}{2}}{2} \]
\[ = \frac{4 + 2a_3}{4} \]

### Step 6: Multiply both sides by 4 to eliminate the fraction
\[ 4a_3 = 4 + 2a_3 \]

### Step 7: Rearrange the equation
Now, we rearrange the equation to isolate \( a_3 \):
\[ 4a_3 - 2a_3 = 4 \]
\[ 2a_3 = 4 \]

### Step 8: Solve for \( a_3 \)
Divide both sides by 2:
\[ a_3 = 2 \]

### Final Answer
Thus, the value of \( a_3 \) is \( 2 \).