Since the known values are 5, the estimation is based on the expansion of Δ5
= (y – 1)5 = 0
∴ Δ5= (y – 1)5
= y5 – 5y4 + 10y3 – 10y2 + 5y1 – y0 = 0
We have to determine the value of y3;
20.1 – 5(23.1) + 10y3 – 10(29.1) + 5(32.2) – 35.4 = 0
So, by simplifying, 10y3 – 260.8 = 0
∴ y3 = 26.08 years
Hence, the probable expectation of life at the age 25 is 26.08 years.
Now expand (y – 1)5 = 0 with change of subscript, keeping coefficients as it is.
y6 – 5y5 + 10y4 – 10y3 + 5y2 – y1 = 0
y6 – 5(20.1) + 10(23.1) – 10(26.08) + 5(29.1)-32.2 = 0
y6– 17 = 0, y6 = 17 years.