If `a_(1)=3 and a_(n)=n+a_(n-1)`, the sum of the first five term is

Options

A. 17
B. 30
C. 42
D. 45

Explanation

To solve the problem step by step, we need to find the first five terms of the sequence defined by the recurrence relation \( a_n = n + a_{n-1} \) with the initial condition \( a_1 = 3 \).

### Step 1: Calculate \( a_2 \)
Using the recurrence relation:
\[ a_2 = 2 + a_{1} \]
Substituting the value of \( a_1 \):
\[ a_2 = 2 + 3 = 5 \]

### Step 2: Calculate \( a_3 \)
Using the recurrence relation:
\[ a_3 = 3 + a_{2} \]
Substituting the value of \( a_2 \):
\[ a_3 = 3 + 5 = 8 \]

### Step 3: Calculate \( a_4 \)
Using the recurrence relation:
\[ a_4 = 4 + a_{3} \]
Substituting the value of \( a_3 \):
\[ a_4 = 4 + 8 = 12 \]

### Step 4: Calculate \( a_5 \)
Using the recurrence relation:
\[ a_5 = 5 + a_{4} \]
Substituting the value of \( a_4 \):
\[ a_5 = 5 + 12 = 17 \]

### Step 5: Calculate the sum of the first five terms
Now we need to find the sum:
\[ S = a_1 + a_2 + a_3 + a_4 + a_5 \]
Substituting the values we found:
\[ S = 3 + 5 + 8 + 12 + 17 \]
Calculating the sum:
\[ S = 3 + 5 = 8 \]
\[ S = 8 + 8 = 16 \]
\[ S = 16 + 12 = 28 \]
\[ S = 28 + 17 = 45 \]

Thus, the sum of the first five terms is \( \boxed{45} \).