Which term of the Arithmetic Progression (AP): 3, 7, 11, ..., 147 is its middle term?

Question

Accepted Answer

#### Step 1: Determine the Total Number of Terms
The given AP has:
- First term ($$a$$) = 3  
- Common difference ($$d$$) = 4  
- Last term ($$l$$) = 147

The formula for the $$n$$-th term of an AP is:  
$$
a_n = a + (n - 1)d
$$
Substitute $$a_n = 147$$:  
$$
147 = 3 + (n - 1) \cdot 4
$$
Simplify:  
$$
147 - 3 = (n - 1) \cdot 4 \implies 144 = (n - 1) \cdot 4
$$
$$
n - 1 = \frac{144}{4} = 36 \implies n = 37
$$

So, the total number of terms ($$n$$) is **37**.

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#### Step 2: Find the Middle Term
The middle term corresponds to the position:  
$$
	ext{Middle Term Position} = \frac{n + 1}{2} = \frac{37 + 1}{2} = 19
$$

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#### Step 3: Verify the Middle Term
The formula for the $$n$$-th term is:  
$$
a_n = a + (n - 1)d
$$
Substitute $$a = 3$$, $$d = 4$$, and $$n = 19$$:  
$$
a_{19} = 3 + (19 - 1) \cdot 4 = 3 + 72 = 75
$$

Thus, the middle term is **75**, and it is the **19th term**.

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Therefore, the correct answer is **(b) 19th**.

Answer

Answer

Answer