Select the number from among the given options that can replace the question mark (?) in the following series. 5, 18, 70, 278, ?
This explanation focuses on determining the missing number in the sequence 5, 18, 70, 278, ?. We need to identify the pattern connecting these numbers to find the value that should replace the question mark (?).
Analyzing the Number Series Pattern
The sequence provided is 5, 18, 70, 278, ?. Our goal is to find the missing term by recognizing the pattern.
Let's examine the relationship between consecutive terms:
- The first term is 5.
- The second term is 18.
- The third term is 70.
- The fourth term is 278.
We can look for a pattern involving multiplication and addition or subtraction.
Consider the operation needed to get from one term to the next:
- To get from 5 to 18: We can try multiplying 5 by a number. $5 \times 3 = 15$ (difference of 3). $5 \times 4 = 20$ (difference of -2).
- Let's test the pattern: Multiply the current term by 4 and subtract 2.
Applying this potential pattern:
- For the second term: $(5 \times 4) - 2 = 20 - 2 = 18$. This matches.
- For the third term: $(18 \times 4) - 2 = 72 - 2 = 70$. This also matches.
- For the fourth term: $(70 \times 4) - 2 = 280 - 2 = 278$. This matches as well.
The pattern identified is consistent across the given terms: Each term is obtained by multiplying the previous term by 4 and then subtracting 2. Mathematically, if $T_n$ represents the $n^{th}$ term, the pattern is $T_{n+1} = (T_n \times 4) - 2$.
Calculating the Missing Number
Now we apply the confirmed pattern to find the number that replaces the question mark.
The last given term in the series is 278.
Using the formula $T_{n+1} = (T_n \times 4) - 2$:
- The next term $= (278 \times 4) - 2$
- First, perform the multiplication: $278 \times 4 = 1112$
- Then, perform the subtraction: $1112 - 2 = 1110$
Therefore, the number that should replace the question mark is 1110.
Conclusion: The Missing Number in the Series
The numerical series follows a specific pattern where each subsequent number is generated by multiplying the previous number by 4 and subtracting 2. Following this rule, the missing number in the sequence 5, 18, 70, 278, ? is calculated to be 1110.
Comparing our result with the provided options:
- 1. 328
- 2. 298
- 3. 592
- 4. 1110
The calculated value of 1110 corresponds to Option 4.
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