The sum of the first 100 terms common to the series
17, 21, 25, 29, 33, …… and 16 , 21 ,26, 31 , 36 …. is

Options

A. 100101
B. 111000
C. 101100
D. 110010

Answer

To find the sum of the first 100 terms common to the series 17, 21, 25, 29, 33, ... and 16, 21, 26, 31, 36, ..., we can follow these steps:

### Step 1: Identify the two arithmetic progressions (APs)
- The first series is: 17, 21, 25, 29, 33, ...
- First term (A1) = 17
- Common difference (D1) = 21 - 17 = 4
- The second series is: 16, 21, 26, 31, 36, ...
- First term (A2) = 16
- Common difference (D2) = 21 - 16 = 5

### Step 2: Find the common terms
The common terms of both series can be found by determining the least common multiple (LCM) of the common differences (D1 and D2).
- LCM of D1 and D2 = LCM(4, 5) = 20

### Step 3: Determine the first common term
The first common term in both series is 21 (as it appears in both).

### Step 4: Formulate the nth term of the common series
The common series can be expressed as:
- First term (A) = 21
- Common difference (D) = 20
- The nth term of the common series can be given by:
\[ a_n = A + (n - 1)D = 21 + (n - 1) \cdot 20 \]

### Step 5: Find the sum of the first 100 terms
The sum of the first n terms of an arithmetic series is given by the formula:
\[ S_n = \frac{n}{2} \cdot (2A + (n - 1)D) \]
For our case:
- n = 100
- A = 21
- D = 20

Substituting these values into the formula:
\[ S_{100} = \frac{100}{2} \cdot (2 \cdot 21 + (100 - 1) \cdot 20) \]
\[ = 50 \cdot (42 + 99 \cdot 20) \]
\[ = 50 \cdot (42 + 1980) \]
\[ = 50 \cdot 2022 \]
\[ = 101100 \]

### Final Answer:
The sum of the first 100 terms common to the series is **101100**.