Which term of the A.P. -2,-7,-12, . . . will be -77 ?

Answer

To find which term of the arithmetic progression (A.P.) -2, -7, -12, ... will be -77, we can use the formula for the nth term of an A.P.:

**Step 1: Identify the first term (a) and the common difference (d).**
- The first term \( a = -2 \).
- To find the common difference \( d \), we subtract the first term from the second term:
\[ d = -7 - (-2) = -7 + 2 = -5. \]

**Step 2: Write the formula for the nth term (Tn) of the A.P.**
- The formula for the nth term is:
\[ T_n = a + (n - 1) \cdot d. \]

**Step 3: Set Tn equal to -77 and substitute the values of a and d.**
- We want to find \( n \) such that:
\[ -77 = -2 + (n - 1)(-5). \]

**Step 4: Simplify the equation.**
- Rearranging the equation gives:
\[ -77 + 2 = (n - 1)(-5). \]
- This simplifies to:
\[ -75 = (n - 1)(-5). \]

**Step 5: Divide both sides by -5.**
- Dividing gives:
\[ 15 = n - 1. \]

**Step 6: Solve for n.**
- Adding 1 to both sides results in:
\[ n = 15 + 1 = 16. \]

Thus, the 16th term of the A.P. is -77.

### Summary of Steps:
1. Identify the first term and common difference.
2. Write the formula for the nth term.
3. Set the nth term equal to -77 and substitute known values.
4. Simplify the equation.
5. Divide to isolate \( n - 1 \).
6. Solve for \( n \).