If the `n^(th)` term of the A.P. `-1, 4, 9, 14,….` is 129, find the value of n.

Answer

To find the value of \( n \) for which the \( n^{th} \) term of the arithmetic progression (A.P.) \(-1, 4, 9, 14, \ldots\) is 129, we will follow these steps:

### Step 1: Identify the first term and common difference
The first term \( a \) of the A.P. is:
\[ a = -1 \]
To find the common difference \( d \), we subtract the first term from the second term:
\[ d = a_2 - a_1 = 4 - (-1) = 4 + 1 = 5 \]

### Step 2: Write the formula for the \( n^{th} \) term
The formula for the \( n^{th} \) term \( a_n \) of an A.P. is given by:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values of \( a \) and \( d \):
\[ a_n = -1 + (n - 1) \cdot 5 \]

### Step 3: Set the \( n^{th} \) term equal to 129
We know that \( a_n = 129 \), so we set up the equation:
\[ -1 + (n - 1) \cdot 5 = 129 \]

### Step 4: Simplify the equation
First, add 1 to both sides:
\[ (n - 1) \cdot 5 = 129 + 1 \]
\[ (n - 1) \cdot 5 = 130 \]

### Step 5: Solve for \( n - 1 \)
Now, divide both sides by 5:
\[ n - 1 = \frac{130}{5} \]
\[ n - 1 = 26 \]

### Step 6: Solve for \( n \)
Finally, add 1 to both sides:
\[ n = 26 + 1 = 27 \]

### Conclusion
Thus, the value of \( n \) is:
\[ \boxed{27} \]

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