Find the number of terms in the sequence `4,7,10,13…….,82.`

Answer

To find the number of terms in the sequence \(4, 7, 10, 13, \ldots, 82\), we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(a\) of the sequence is \(4\) and the common difference \(d\) can be calculated as follows:
\[ d = 7 - 4 = 3 \]

### Step 2: Identify the last term
The last term \(T_n\) of the sequence is given as \(82\).

### Step 3: Use the formula for the \(n\)-th term of an arithmetic sequence
The formula for the \(n\)-th term of an arithmetic sequence is:
\[ T_n = a + (n - 1) \cdot d \]
Substituting the known values into the formula:
\[ 82 = 4 + (n - 1) \cdot 3 \]

### Step 4: Simplify the equation
First, subtract \(4\) from both sides:
\[ 82 - 4 = (n - 1) \cdot 3 \]
\[ 78 = (n - 1) \cdot 3 \]

### Step 5: Divide both sides by \(3\)
\[ n - 1 = \frac{78}{3} \]
\[ n - 1 = 26 \]

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

### Conclusion
The number of terms in the sequence is \(27\).

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