nth term of A.P. 5,10, 15, 20,……….n terms and nth term of A.P. 15, 30, 45, 60,…………n terms are same.

Answer

To determine whether the nth terms of the two given arithmetic progressions (A.P.s) are the same, we will follow these steps:

### Step 1: Identify the first A.P. and its parameters
The first A.P. is 5, 10, 15, 20, ...
- First term (a1) = 5
- Common difference (d1) = second term - first term = 10 - 5 = 5

### Step 2: Write the formula for the nth term of the first A.P.
The formula for the nth term of an A.P. is given by:
\[ a_n = a + (n - 1) \times d \]
For the first A.P.:
\[ a_n = 5 + (n - 1) \times 5 \]

### Step 3: Simplify the nth term of the first A.P.
Now, we simplify the expression:
\[ a_n = 5 + 5(n - 1) \]
\[ a_n = 5 + 5n - 5 \]
\[ a_n = 5n \]

### Step 4: Identify the second A.P. and its parameters
The second A.P. is 15, 30, 45, 60, ...
- First term (a2) = 15
- Common difference (d2) = second term - first term = 30 - 15 = 15

### Step 5: Write the formula for the nth term of the second A.P.
For the second A.P.:
\[ a_n = 15 + (n - 1) \times 15 \]

### Step 6: Simplify the nth term of the second A.P.
Now, we simplify the expression:
\[ a_n = 15 + 15(n - 1) \]
\[ a_n = 15 + 15n - 15 \]
\[ a_n = 15n \]

### Step 7: Compare the nth terms of both A.P.s
Now we have:
- nth term of the first A.P. = \( 5n \)
- nth term of the second A.P. = \( 15n \)

### Step 8: Determine if the nth terms are the same
To check if they are the same, we set them equal:
\[ 5n = 15n \]

### Step 9: Solve the equation
Subtract \( 5n \) from both sides:
\[ 0 = 10n \]
This implies \( n = 0 \).

Since \( n \) cannot be zero in the context of counting terms in an A.P., we conclude that the nth terms of both A.P.s are not the same for any positive integer \( n \).

### Conclusion
The statement that the nth terms of both A.P.s are the same is **false**.

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