IF `a_n=1/(n+1)+1` then find the value of `a_1+a_3+a_5`

Options

A. `17/12`
B. `27/12`
C. `37/12`
D. `47/12`

Answer

To solve the problem, we need to find the values of \( a_1 \), \( a_3 \), and \( a_5 \) using the formula given for \( a_n \):

\[ a_n = \frac{1}{n + 1} + 1 \]

### Step 1: Calculate \( a_1 \)

Substituting \( n = 1 \):

\[ a_1 = \frac{1}{1 + 1} + 1 = \frac{1}{2} + 1 = \frac{1}{2} + \frac{2}{2} = \frac{3}{2} \]

### Step 2: Calculate \( a_3 \)

Substituting \( n = 3 \):

\[ a_3 = \frac{1}{3 + 1} + 1 = \frac{1}{4} + 1 = \frac{1}{4} + \frac{4}{4} = \frac{5}{4} \]

### Step 3: Calculate \( a_5 \)

Substituting \( n = 5 \):

\[ a_5 = \frac{1}{5 + 1} + 1 = \frac{1}{6} + 1 = \frac{1}{6} + \frac{6}{6} = \frac{7}{6} \]

### Step 4: Calculate \( a_1 + a_3 + a_5 \)

Now, we add \( a_1 \), \( a_3 \), and \( a_5 \):

\[ a_1 + a_3 + a_5 = \frac{3}{2} + \frac{5}{4} + \frac{7}{6} \]

To add these fractions, we need a common denominator. The least common multiple (LCM) of 2, 4, and 6 is 12.

### Step 5: Convert each fraction to have a denominator of 12

\[ \frac{3}{2} = \frac{3 \times 6}{2 \times 6} = \frac{18}{12} \]

\[ \frac{5}{4} = \frac{5 \times 3}{4 \times 3} = \frac{15}{12} \]

\[ \frac{7}{6} = \frac{7 \times 2}{6 \times 2} = \frac{14}{12} \]

### Step 6: Add the fractions

Now we can add them:

\[ a_1 + a_3 + a_5 = \frac{18}{12} + \frac{15}{12} + \frac{14}{12} = \frac{18 + 15 + 14}{12} = \frac{47}{12} \]

### Final Answer

Thus, the value of \( a_1 + a_3 + a_5 \) is:

\[ \frac{47}{12} \]

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