How many terms are common in two arithmetic progression 1,4,7,10,…` upto 63 terms and 3,7,11,15 ,… upto 47 terms :

Options

A. 12
B. 16
C. 15
D. none of these

Answer

To find how many terms are common in the two arithmetic progressions (APs) given, we can follow these steps:

### Step 1: Identify the first AP and its properties
The first AP is given as:
1, 4, 7, 10, ..., up to 63 terms.

- First term (a1) = 1
- Common difference (d1) = 4 - 1 = 3
- Number of terms (n1) = 63

To find the 63rd term (a63) of this AP, we use the formula for the nth term of an AP:
\[ a_n = a + (n - 1) \cdot d \]
Substituting the values:
\[ a_{63} = 1 + (63 - 1) \cdot 3 \]
\[ a_{63} = 1 + 62 \cdot 3 \]
\[ a_{63} = 1 + 186 = 187 \]

### Step 2: Identify the second AP and its properties
The second AP is given as:
3, 7, 11, 15, ..., up to 47 terms.

- First term (a2) = 3
- Common difference (d2) = 7 - 3 = 4
- Number of terms (n2) = 47

To find the 47th term (a47) of this AP, we use the same formula:
\[ a_{47} = 3 + (47 - 1) \cdot 4 \]
\[ a_{47} = 3 + 46 \cdot 4 \]
\[ a_{47} = 3 + 184 = 187 \]

### Step 3: Determine the common terms in both APs
Now we need to find the common terms in both APs. The first common term is 7 (the first term of the first AP after 1). The last common term we found is 187.

### Step 4: Find the common difference
The common difference between the two APs is the least common multiple (LCM) of their common differences:
- d1 = 3
- d2 = 4

The LCM of 3 and 4 is 12.

### Step 5: Form the new AP of common terms
The common terms form a new AP starting from 7 with a common difference of 12:
7, 19, 31, ..., up to 187.

### Step 6: Find the number of terms in the new AP
To find how many terms are in this new AP, we use the nth term formula again:
Let the number of terms be n.
\[ a_n = 7 + (n - 1) \cdot 12 = 187 \]
Rearranging gives:
\[ 187 - 7 = (n - 1) \cdot 12 \]
\[ 180 = (n - 1) \cdot 12 \]
\[ n - 1 = \frac{180}{12} = 15 \]
\[ n = 15 + 1 = 16 \]

### Conclusion
The total number of common terms in the two arithmetic progressions is **16**.

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