A sequence S is defined as follows : `S_(n)=(S_(n+1)+S_(n-1))/(2)`. If `S_(1)=15 and S_(4)=10.5,` What is `S_(2)`?

Answer

To solve the problem, we need to find the value of \( S_2 \) given the sequence defined by the equation:

\[ S_n = \frac{S_{n+1} + S_{n-1}}{2} \]

We know that \( S_1 = 15 \) and \( S_4 = 10.5 \).

### Step 1: Express \( S_2 \) in terms of \( S_1 \) and \( S_3 \)

Using the sequence definition for \( n = 2 \):

\[ S_2 = \frac{S_3 + S_1}{2} \]

Substituting \( S_1 = 15 \):

\[ S_2 = \frac{S_3 + 15}{2} \tag{1} \]

### Step 2: Express \( S_3 \) in terms of \( S_2 \) and \( S_4 \)

Now, using the sequence definition for \( n = 3 \):

\[ S_3 = \frac{S_4 + S_2}{2} \]

Substituting \( S_4 = 10.5 \):

\[ S_3 = \frac{10.5 + S_2}{2} \tag{2} \]

### Step 3: Substitute \( S_3 \) from equation (2) into equation (1)

Now we can substitute the expression for \( S_3 \) from equation (2) into equation (1):

\[ S_2 = \frac{\left(\frac{10.5 + S_2}{2}\right) + 15}{2} \]

### Step 4: Simplify the equation

Multiply both sides by 2 to eliminate the fraction:

\[ 2S_2 = \frac{10.5 + S_2}{2} + 15 \]

Now multiply everything by 2 again to clear the remaining fraction:

\[ 4S_2 = 10.5 + S_2 + 30 \]

Combine like terms:

\[ 4S_2 - S_2 = 40.5 \]

This simplifies to:

\[ 3S_2 = 40.5 \]

### Step 5: Solve for \( S_2 \)

Now divide both sides by 3:

\[ S_2 = \frac{40.5}{3} = 13.5 \]

Thus, the value of \( S_2 \) is:

\[ \boxed{13.5} \]