If average of a 10 term A.P is 112.5% more than its first term then second term of A.P is what % of the sum of the series?

Options

A. `2(5)/17%`
B. `4(10)/17%`
C. `5(15)/17%`
D. `3(16)/17%`

Answer

To solve the problem step by step, we will follow the reasoning provided in the video transcript:

### Step 1: Understand the Average of the A.P.
The average of a 10-term Arithmetic Progression (A.P.) is given to be 112.5% more than its first term.

Let the first term of the A.P. be \( a \).

### Step 2: Calculate the Average
The average of the first 10 terms of the A.P. can be calculated using the formula:
\[ \text{Average} = \frac{\text{Sum of terms}}{\text{Number of terms}} = \frac{S_{10}}{10} \]
Where \( S_{10} \) is the sum of the first 10 terms.

### Step 3: Express the Sum of the A.P.
The sum of the first 10 terms of an A.P. can also be expressed as:
\[ S_{10} = \frac{n}{2} \times (\text{first term} + \text{last term}) = \frac{10}{2} \times (a + (a + 9d)) = 5 \times (2a + 9d) = 10a + 45d \]

### Step 4: Set Up the Equation for Average
Thus, the average becomes:
\[ \text{Average} = \frac{10a + 45d}{10} = a + 4.5d \]
According to the problem, this average is 112.5% more than the first term \( a \). Therefore:
\[ a + 4.5d = a + 1.125a = 2.125a \]

### Step 5: Solve for \( d \)
Now, we can set up the equation:
\[ 4.5d = 2.125a - a \]
\[ 4.5d = 1.125a \]
Dividing both sides by 4.5 gives:
\[ d = \frac{1.125a}{4.5} = \frac{1.125}{4.5}a = \frac{1}{4}a \]

### Step 6: Calculate the Total Sum of the A.P.
Now, substituting \( d \) back into the sum of the A.P.:
\[ S_{10} = 10a + 45d = 10a + 45 \left(\frac{1}{4}a\right) = 10a + \frac{45}{4}a = 10a + 11.25a = 21.25a \]

### Step 7: Find the Second Term
The second term of the A.P. is:
\[ \text{Second term} = a + d = a + \frac{1}{4}a = \frac{5}{4}a \]

### Step 8: Calculate the Percentage of the Second Term with Respect to the Total Sum
Now, we need to find what percentage the second term is of the total sum:
\[ \text{Percentage} = \left(\frac{\text{Second term}}{S_{10}}\right) \times 100 = \left(\frac{\frac{5}{4}a}{21.25a}\right) \times 100 \]
The \( a \) cancels out:
\[ = \left(\frac{\frac{5}{4}}{21.25}\right) \times 100 = \left(\frac{5}{4 \times 21.25}\right) \times 100 = \left(\frac{5}{85}\right) \times 100 = \frac{500}{85} \]
Calculating \( \frac{500}{85} \) gives approximately \( 5.88\% \).

### Step 9: Simplify the Percentage
To express this as a fraction:
\[ \frac{500}{85} = \frac{100}{17} \]
Thus, the percentage is:
\[ \frac{100}{17} \approx 5.88\% \]

### Final Answer
The second term of the A.P. is \( \frac{100}{17}\% \) of the sum of the series.