Series Pattern Analysis

The question asks us to identify the missing number in the given sequence: $203 \ 189 \ 170 \ 146 \ 117 \ ?$. To find the missing term, we need to carefully examine the relationship between the numbers in the sequence and find a consistent pattern.

Differences Between Terms

Let's start by calculating the difference between each consecutive pair of numbers in the series:

• The difference between the first and second term is $203 - 189 = 14$.
• The difference between the second and third term is $189 - 170 = 19$.
• The difference between the third and fourth term is $170 - 146 = 24$.
• The difference between the fourth and fifth term is $146 - 117 = 29$.

So, the sequence of differences is $14, 19, 24, 29$.

Pattern in Differences

Now, we look for a pattern within this sequence of differences ($14, 19, 24, 29$):

• The difference between $19$ and $14$ is $19 - 14 = 5$.
• The difference between $24$ and $19$ is $24 - 19 = 5$.
• The difference between $29$ and $24$ is $29 - 24 = 5$.

We can see a clear pattern here: each difference is $5$ greater than the previous difference.

Next Term Calculation

To find the next difference in the sequence, we add $5$ to the last calculated difference ($29$):

Next difference = $29 + 5 = 34$.

Now, to find the missing term (the one represented by '?'), we subtract this next difference ($34$) from the last known term in the original sequence ($117$):

Missing Term = $117 - 34$.

Calculating this subtraction gives us:

Missing Term = $83$.

Series Conclusion

By analyzing the differences between terms and identifying the increasing pattern, we determined that the number $83$ should logically follow $117$ in the given sequence.