It the sum of first `10` terms of an arithmetic progression with first term `p` and common difference `q`, is `4` times the sum of the first `5` terms, then what is the ratio `p : q` ?

Options

A. `1 : 2`
B. `1 : 4`
C. `2 : 1`
D. `4 : 1`

Explanation

To find the ratio \( p : q \) given that the sum of the first 10 terms of an arithmetic progression (AP) with first term \( p \) and common difference \( q \) is 4 times the sum of the first 5 terms, we can follow these steps:

### Step 1: Write the formula for the sum of the first \( n \) terms of an AP

The formula for the sum of the first \( n \) terms \( S_n \) of an arithmetic progression is given by:

\[ S_n = \frac{n}{2} \left(2a + (n-1)d\right) \]

where:
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the number of terms.

### Step 2: Calculate the sum of the first 10 terms

For the first 10 terms, we have:

\[ S_{10} = \frac{10}{2} \left(2p + (10-1)q\right) \]

This simplifies to:

\[ S_{10} = 5 \left(2p + 9q\right) = 10p + 45q \]

### Step 3: Calculate the sum of the first 5 terms

For the first 5 terms, we have:

\[ S_{5} = \frac{5}{2} \left(2p + (5-1)q\right) \]

This simplifies to:

\[ S_{5} = \frac{5}{2} \left(2p + 4q\right) = 5p + 10q \]

### Step 4: Set up the equation based on the problem statement

According to the problem, the sum of the first 10 terms is 4 times the sum of the first 5 terms:

\[ S_{10} = 4 \times S_{5} \]

Substituting the expressions we found:

\[ 10p + 45q = 4(5p + 10q) \]

### Step 5: Expand and simplify the equation

Expanding the right side:

\[ 10p + 45q = 20p + 40q \]

Now, rearranging the equation gives:

\[ 10p + 45q - 20p - 40q = 0 \]

This simplifies to:

\[ -10p + 5q = 0 \]

### Step 6: Solve for the ratio \( p : q \)

Rearranging gives:

\[ 10p = 5q \]

Dividing both sides by \( 5q \):

\[ \frac{p}{q} = \frac{5}{10} = \frac{1}{2} \]

Thus, the ratio \( p : q \) is:

\[ p : q = 1 : 2 \]

### Final Answer

The ratio \( p : q \) is \( 1 : 2 \).

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