Which term of the series 3, 8, 13, 18, ........ is 78 ?

Question

Accepted Answer

To determine which term of the series 3, 8, 13, 18, ... is 78, we first need to identify the pattern of the series. This series is an arithmetic sequence where each term increases by a common difference.

1. **Identify the common difference:**
   The difference between consecutive terms is:
   - 8 - 3 = 5
   - 13 - 8 = 5
   - 18 - 13 = 5

Therefore, the common difference (d) is 5.

2. **Use the formula for the nth term of an arithmetic sequence:**
   The formula for the nth term (a_n) of an arithmetic sequence is:
   $$
   a_n = a_1 + (n - 1) \times d
   $$
   where $a_1$ is the first term and $d$ is the common difference.

3. **Substitute the known values to find n:**
   We know $a_1 = 3$, $d = 5$, and we want to find n such that $a_n = 78$.

$$
   78 = 3 + (n - 1) \times 5
   $$

Simplifying the equation:
   $$
   78 = 3 + 5n - 5
   $$
   $$
   78 = 5n - 2
   $$
   $$
   80 = 5n
   $$
   $$
   n = \frac{80}{5} = 16
   $$

Therefore, the 16th term of the series is 78.

4. **Explanation of why Option 3 is correct:**
   Option 3 states that the 16th term is 78, which matches our calculation. Thus, Option 3 is the correct answer.

5. **Explanation of why other options are incorrect:**
   - **Option 1 (18):** If the 18th term were 78, substituting n = 18 in the formula would give:
     $$
     a_{18} = 3 + (18 - 1) \times 5 = 3 + 85 = 88
     $$
     This does not equal 78, so Option 1 is incorrect.

- **Option 2 (76):** If the 76th term were 78, substituting n = 76 in the formula would give:
     $$
     a_{76} = 3 + (76 - 1) \times 5 = 3 + 375 = 378
     $$
     This does not equal 78, so Option 2 is incorrect.

- **Option 4 (74):** If the 74th term were 78, substituting n = 74 in the formula would give:
     $$
     a_{74} = 3 + (74 - 1) \times 5 = 3 + 365 = 368
     $$
     This does not equal 78, so Option 4 is incorrect.

In conclusion, Option 3 is the correct answer as the 16th term of the series is indeed 78.