Find the sum of first five terms in the sequence
`1/(1xx4)+1/(4xx7)+1/(7xx10)+.....`

Options

A. `1/32`
B. `7/16`
C. `5/16`
D. `1/210`

Answer

To find the sum of the first five terms in the sequence \( \frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10} + \ldots \), we can follow these steps:

### Step 1: Identify the pattern in the denominators
The denominators of the terms in the sequence are products of numbers that increase by 3 each time. The first term has \( 1 \) and \( 4 \), the second term has \( 4 \) and \( 7 \), and the third term has \( 7 \) and \( 10 \). The next terms will follow this pattern:
- The first term is \( 1 \times 4 \)
- The second term is \( 4 \times 7 \)
- The third term is \( 7 \times 10 \)
- The fourth term will be \( 10 \times 13 \)
- The fifth term will be \( 13 \times 16 \)

### Step 2: Write down the first five terms
Now we can write down the first five terms:
1. \( \frac{1}{1 \times 4} = \frac{1}{4} \)
2. \( \frac{1}{4 \times 7} = \frac{1}{28} \)
3. \( \frac{1}{7 \times 10} = \frac{1}{70} \)
4. \( \frac{1}{10 \times 13} = \frac{1}{130} \)
5. \( \frac{1}{13 \times 16} = \frac{1}{208} \)

### Step 3: Find a common denominator
To add these fractions, we need a common denominator. The least common multiple (LCM) of \( 4, 28, 70, 130, \) and \( 208 \) can be calculated. However, for simplicity, we can calculate the sum directly using a calculator or by finding the LCM step by step.

### Step 4: Calculate the sum
Now we can add the fractions:
\[ S = \frac{1}{4} + \frac{1}{28} + \frac{1}{70} + \frac{1}{130} + \frac{1}{208} \]

Calculating each term with a common denominator (which can be found to be \( 840 \)):
1. \( \frac{1}{4} = \frac{210}{840} \)
2. \( \frac{1}{28} = \frac{30}{840} \)
3. \( \frac{1}{70} = \frac{12}{840} \)
4. \( \frac{1}{130} = \frac{6}{840} \)
5. \( \frac{1}{208} = \frac{4}{840} \)

Now, summing these:
\[ S = \frac{210 + 30 + 12 + 6 + 4}{840} = \frac{262}{840} \]

### Step 5: Simplify the fraction
Now we simplify \( \frac{262}{840} \):
\[ \frac{262 \div 2}{840 \div 2} = \frac{131}{420} \]

### Final Answer
Thus, the sum of the first five terms in the sequence is \( \frac{131}{420} \).