Find the first six terms of the sequence whose first temr is 1 and whose (n+1) th term is obtained by adding n to the nth term.

Answer

To find the first six terms of the sequence defined by the given conditions, we will follow these steps:

1. **Identify the first term**: The first term \( a_1 \) is given as 1.
\[ a_1 = 1 \]

2. **Define the recursive formula**: The \( (n+1) \)th term is obtained by adding \( n \) to the \( n \)th term. This can be expressed as:
\[ a_{n+1} = a_n + n \]

3. **Calculate the second term \( a_2 \)**: For \( n = 1 \):
\[ a_2 = a_1 + 1 = 1 + 1 = 2 \]

4. **Calculate the third term \( a_3 \)**: For \( n = 2 \):
\[ a_3 = a_2 + 2 = 2 + 2 = 4 \]

5. **Calculate the fourth term \( a_4 \)**: For \( n = 3 \):
\[ a_4 = a_3 + 3 = 4 + 3 = 7 \]

6. **Calculate the fifth term \( a_5 \)**: For \( n = 4 \):
\[ a_5 = a_4 + 4 = 7 + 4 = 11 \]

7. **Calculate the sixth term \( a_6 \)**: For \( n = 5 \):
\[ a_6 = a_5 + 5 = 11 + 5 = 16 \]

8. **List the first six terms**: The first six terms of the sequence are:
\[ a_1 = 1, \quad a_2 = 2, \quad a_3 = 4, \quad a_4 = 7, \quad a_5 = 11, \quad a_6 = 16 \]

Thus, the first six terms of the sequence are:
\[ 1, 2, 4, 7, 11, 16 \]