The average of all 3-digit terms in the arithmetic progression 38, 55, 72, ..., is

Question

Accepted Answer

#### Step 1: Identify Key Parameters of the AP  
The given AP has:  
- First term ($$a$$) = 38  
- Common difference ($$d$$) = 17

#### Step 2: Find the Smallest and Largest 3-Digit Terms  
The general term of the AP is given by:  
$$
T_n = a + (n-1)d.
$$

##### Smallest 3-digit term:  
Set $$T_n \geq 100$$:  
$$
38 + (n-1) 	imes 17 \geq 100.
$$  
Simplify:  
$$
(n-1) 	imes 17 \geq 62 \implies n-1 \geq \frac{62}{17} \approx 3.65.
$$  
Thus, $$n = 5$$. The smallest term is:  
$$
T_5 = 38 + (5-1) 	imes 17 = 106.
$$

##### Largest 3-digit term:  
Set $$T_n \leq 999$$:  
$$
38 + (n-1) 	imes 17 \leq 999.
$$  
Simplify:  
$$
(n-1) 	imes 17 \leq 961 \implies n-1 \leq \frac{961}{17} \approx 56.53.
$$  
Thus, $$n = 57$$. The largest term is:  
$$
T_{57} = 38 + (57-1) 	imes 17 = 990.
$$

#### Step 3: Calculate the Average of All Terms  
The terms between $$106$$ and $$990$$ form an AP with first term $$106$$, last term $$990$$, and common difference $$d = 17$$.

##### Number of terms in this sequence:  
Use the formula for the nth term:  
$$
T_n = a + (n-1)d.
$$  
Substitute $$T_n = 990, a = 106, d = 17$$:  
$$
990 = 106 + (n-1) 	imes 17.
$$  
Simplify:  
$$
884 = (n-1) 	imes 17 \implies n-1 = \frac{884}{17} = 52.
$$  
Thus, $$n = 53.$$

##### Average formula for an AP:  
The average of an AP is given by the mean of its first and last terms:  
$$
	ext{Average} = \frac{	ext{First Term} + 	ext{Last Term}}{2}.
$$  
Substitute values:  
$$
	ext{Average} = \frac{106 + 990}{2} = \frac{1096}{2} = 548.
$$

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### Final Answer:  
The average of all the three-digit terms in the arithmetic progression is **548**.