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 common difference
The first term \(a_1\) of the AP is \(4\) and the second term \(a_2\) is \(8\). The common difference \(d\) can be calculated as:
\[ d = a_2 - a_1 = 8 - 4 = 4 \]

### Step 2: Identify the last term
The last term \(l\) of the AP is given as \(64\).

### Step 3: Find the number of terms \(n\)
We can use the formula for the \(n\)-th term of an AP:
\[ T_n = a + (n-1)d \]
Setting \(T_n\) equal to the last term \(l\):
\[ 64 = 4 + (n-1) \cdot 4 \]
Subtract \(4\) from both sides:
\[ 60 = (n-1) \cdot 4 \]
Now, divide both sides by \(4\):
\[ n - 1 = 15 \]
Adding \(1\) to both sides gives:
\[ n = 16 \]

### Step 4: Use the sum formula for an AP
The sum \(S_n\) of the first \(n\) terms of an AP can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we have:
\[ S_{16} = \frac{16}{2} \cdot (4 + 64) \]
Calculating inside the parentheses:
\[ S_{16} = 8 \cdot 68 \]
Now, multiply:
\[ S_{16} = 544 \]

### Final Answer
The sum of the given arithmetic progression \(4 + 8 + 12 + \ldots + 64\) is \(544\).

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