Find the missing value '?' in the following series :
13, 34, 74, ?, 290

Options

A. 168
B. 170
C. 172
D. 174

Answer

To find the missing value '?' in the series: 13, 34, 74, ?, 290, we will look for a pattern in the numbers.

### Step 1: Identify the pattern
Let's analyze the given numbers:

- The first number is 13.
- The second number is 34.
- The third number is 74.
- The fifth number is 290.

We need to find the fourth number.

### Step 2: Break down the numbers into squares
We can observe the numbers and try to express them as sums of squares of consecutive prime numbers.

1. **13** can be expressed as:
\[ 2^2 + 3^2 = 4 + 9 = 13 \]

2. **34** can be expressed as:
\[ 3^2 + 5^2 = 9 + 25 = 34 \]

3. **74** can be expressed as:
\[ 5^2 + 7^2 = 25 + 49 = 74 \]

### Step 3: Find the missing number
Following the pattern, the next number should be the sum of the squares of the next two consecutive prime numbers, which are 7 and 11.

4. Calculate:
\[ 7^2 + 11^2 = 49 + 121 = 170 \]

### Step 4: Verify the last number
Now let's check the last number in the series, 290, to ensure the pattern holds:
- The next consecutive prime numbers after 11 are 11 and 13.
- Calculate:
\[ 11^2 + 13^2 = 121 + 169 = 290 \]

### Conclusion
Thus, the missing value '?' in the series is:
\[ \boxed{170} \]