The sum of the first 10 terms common to the series 17, 21, 25, ... and 16, 21, 26,31 ,`cdots` is

Options

A. 1100
B. 1010
C. 1110
D. 1200

Answer

To find the sum of the first 10 terms common to the series 17, 21, 25, ... and 16, 21, 26, 31, ..., we can follow these steps:

### Step 1: Identify the two series
The first series is:
- \( S_1: 17, 21, 25, \ldots \)

The second series is:
- \( S_2: 16, 21, 26, 31, \ldots \)

### Step 2: Determine the common terms
The first common term is clearly 21 (as it appears in both series).

### Step 3: Identify the common differences
- For the first series \( S_1 \):
- First term \( a_1 = 17 \)
- Common difference \( d_1 = 4 \)

- For the second series \( S_2 \):
- First term \( a_2 = 16 \)
- Common difference \( d_2 = 5 \)

### Step 4: Find the common difference of the combined series
To find the common difference of the series formed by the common terms, we take the least common multiple (LCM) of the two common differences:
- \( \text{LCM}(4, 5) = 20 \)

### Step 5: Write the formula for the common terms
The common terms can be expressed as:
- First common term \( a = 21 \)
- Common difference \( d = 20 \)

### Step 6: Use the formula for the sum of an arithmetic series
The sum \( S_n \) of the first \( n \) terms of an arithmetic series is given by:
\[ S_n = \frac{n}{2} \times (2a + (n-1)d) \]
Where:
- \( n = 10 \) (the number of terms)
- \( a = 21 \) (the first common term)
- \( d = 20 \) (the common difference)

### Step 7: Substitute the values into the formula
Substituting the values into the formula:
\[ S_{10} = \frac{10}{2} \times (2 \times 21 + (10-1) \times 20) \]
\[ = 5 \times (42 + 9 \times 20) \]
\[ = 5 \times (42 + 180) \]
\[ = 5 \times 222 \]
\[ = 1110 \]

### Conclusion
The sum of the first 10 terms common to the two series is **1110**.

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