Answer :

To find the value of [tex]\( a \)[/tex], given the mean [tex]\(\bar{X} = 20\)[/tex], the sum of the products of frequencies and their corresponding values [tex]\( \sum fx = 420 \)[/tex], and the total number of observations [tex]\( N = 18 + 2a \)[/tex], we can follow these steps:

1. Understand the given formula for the mean:
The formula for the mean ([tex]\(\bar{X}\)[/tex]) of a frequency distribution is:
[tex]\[ \bar{X} = \frac{\sum fx}{N} \][/tex]

2. Substitute the given values:
Given [tex]\(\bar{X} = 20\)[/tex] and [tex]\(\sum fx = 420\)[/tex], we substitute these into the formula:
[tex]\[ 20 = \frac{420}{18 + 2a} \][/tex]

3. Solve for [tex]\( N \)[/tex]:
Multiply both sides of the equation by [tex]\( (18 + 2a) \)[/tex] to isolate the total number of observations:
[tex]\[ 20 (18 + 2a) = 420 \][/tex]

4. Expand the equation:
[tex]\[ 20 \times 18 + 20 \times 2a = 420 \][/tex]

Simplify the terms:
[tex]\[ 360 + 40a = 420 \][/tex]

5. Isolate the term with [tex]\( a \)[/tex]:
Subtract 360 from both sides of the equation:
[tex]\[ 40a = 420 - 360 \][/tex]

Simplify the right-hand side:
[tex]\[ 40a = 60 \][/tex]

6. Solve for [tex]\( a \)[/tex]:
Divide both sides by 40:
[tex]\[ a = \frac{60}{40} = 1.5 \][/tex]

So the value of [tex]\( a \)[/tex] is [tex]\( 1.5 \)[/tex].