Introduction:
In mathematics, proportion is a statement that two ratios are equal. It is often used to compare quantities and express relationships between numbers. In this question, we are given a set of numbers (14, 16, 35, 42) and we need to find the fourth term for which these numbers will be in proportion.

Solution:
To find the fourth term, we first need to determine if the given numbers are in proportion. For numbers to be in proportion, the ratios of consecutive terms should be equal.

Step 1: Calculate Ratios:
Let's calculate the ratios of consecutive terms to check if they are equal:

Ratio of 14 and 16: 14/16 = 0.875
Ratio of 16 and 35: 16/35 = 0.457
Ratio of 35 and 42: 35/42 = 0.833

As we can see, the ratios are not equal. Therefore, the given numbers (14, 16, 35, 42) are not in proportion.

Step 2: Finding the Fourth Term:
To find the fourth term for which the given numbers will be in proportion, we can use the concept of cross-multiplication. In a proportion, the product of the means (middle terms) is equal to the product of the extremes (outer terms).

Let's assume the fourth term as 'x'. Therefore, the proportion will be:

14/16 = 35/42 = x/42

Step 3: Cross-multiplication:
Using cross-multiplication, we can solve for 'x':

(14 * 42) = (16 * x)
588 = 16x

Step 4: Solving for 'x':
To find the value of 'x', we can divide both sides of the equation by 16:

x = 588/16
x = 36.75

Therefore, the fourth term for which the numbers (14, 16, 35, 42) will be in proportion is 36.75.

Conclusion:
The given numbers (14, 16, 35, 42) are not in proportion. However, by using the concept of cross-multiplication, we can find the fourth term for which they will be in proportion. In this case, the fourth term is 36.75.