First, find the sales of the fifth firm.
Sum of the first four firms is $2,170,000 and this represents 80% of the market.
Letting X represent the total market, and writing 80% as .80:
Thus: .80 · X = 2,170,000 ---> X = 2, 712, 500
The fifth firm therefore has value: 2,712,500 - 2,170,000 = 542,000
To find the index, sum the square of each of the percentages:
800 000 / 2 712 500 = 29.5%
700 000 / 2 712 500 = 25.8%
440 000 / 2 712 500 = 16.2%
230 000 / 2 712 500 = 8.5%
542 000 / 2 712 500 = 20.0%
Index = (.295)² + (.258)² + (.162)² + (.085)² + (.200)² = 0.227