Find the sum of given Arithmetic Progression `4+8+ 12+....+ 64`

Options

A. 464
B. 540
C. 544
D. 646

Explanation

To find the sum of the given Arithmetic Progression (AP) \(4 + 8 + 12 + \ldots + 64\), we can follow these steps:

### Step 1: Identify the first term and the common difference
The first term \(a\) of the AP is \(4\). The common difference \(d\) can be calculated as:
\[ d = 8 - 4 = 4 \]

### Step 2: Find the number of terms \(n\)
We know that the last term \(l\) is \(64\). The formula for the \(n\)-th term of an AP is given by:
\[ l = a + (n - 1)d \]
Substituting the known values:
\[ 64 = 4 + (n - 1) \cdot 4 \]
Now, simplify the equation:
\[ 64 - 4 = (n - 1) \cdot 4 \]
\[ 60 = (n - 1) \cdot 4 \]
Dividing both sides by \(4\):
\[ 15 = n - 1 \]
Adding \(1\) to both sides gives:
\[ n = 16 \]

### Step 3: Use the formula for the sum of the first \(n\) terms of an AP
The formula for the sum \(S_n\) of the first \(n\) terms of an AP is:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we have:
\[ S_{16} = \frac{16}{2} \cdot (4 + 64) \]
Calculating further:
\[ S_{16} = 8 \cdot 68 \]
\[ S_{16} = 544 \]

### Conclusion
The sum of the given Arithmetic Progression \(4 + 8 + 12 + \ldots + 64\) is \(544\).

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