Solving the Number Series Problem

The question asks us to identify the pattern in the given number series and find the missing term represented by the question mark (?). The series is: 3, 8, 18, 38, ?, 158, 318.

Analyzing the Series Pattern

Let's examine the relationship between consecutive numbers in the series:

• The first term is 3.
• The second term is 8.
• The third term is 18.
• The fourth term is 38.
• The term after the missing one is 158.
• The last term is 318.

We need to find a consistent rule that connects these numbers.

Let's test a common pattern: multiplying the previous term by a number and adding or subtracting another number.

• From 3 to 8: Maybe multiply by 2 and add 2? Let's check: $(3 \times 2) + 2 = 6 + 2 = 8$. This works.
• From 8 to 18: Using the same rule: $(8 \times 2) + 2 = 16 + 2 = 18$. This also works.
• From 18 to 38: Applying the rule again: $(18 \times 2) + 2 = 36 + 2 = 38$. The pattern holds.

So, the identified pattern is: Multiply the current number by 2 and add 2 to get the next number.

Calculating the Missing Term

Now, we apply this rule to find the missing term (?) which comes after 38:

Missing Term $= (38 \times 2) + 2$

Missing Term $= 76 + 2$

Missing Term $= 78$

Verifying the Pattern

Let's confirm if this calculated term (78) fits the rest of the series:

• The term after 78 should be 158. Let's check using our rule: $(78 \times 2) + 2 = 156 + 2 = 158$. It matches perfectly.
• The final term is 318. Let's check the step from 158 to 318: $(158 \times 2) + 2 = 316 + 2 = 318$. This also matches.

Since the pattern $(n \times 2) + 2$ consistently works for all the given terms, the missing number is indeed 78.

Conclusion

The missing number in the series 3, 8, 18, 38, ?, 158, 318 is 78.