Question

The population of a city in a censes taken once in 10 years is given below. Estimate the population in the year 1955.

Year	1951	1961	1971	1981
Population in lakhs	35	42	58	84

Answer 1

Let the year be x and population be y.

To find the population for the year 1955.

i.e. The value of y at x = 1955

Since the value of y is required near the beginning of the table

We use the Newton’s forward interpolation formula.

`y_((x  =  x_+  "nh")) = y_0 + "n"/(1!) Deltay_0 + ("n"("n" - 1))/(2!) Delta^2y_0 + ("n"("n" - 1)("n" - 2))/(3!) Delta^3y_0 + .........`

To find y at x = 1955

∴ x₀ + nh = 1955, x₀ = 1951, h = 10

⇒ 1951 + n(10) = 1955

10n = 1955 – 1951

⇒ 10n = 4

n = `4/10` = 0.4

x	y	`Deltay`	`Delta^2y`	`Delta^3y`
1951	35
		7
1961	42		9
		16		1
1971	58		10
		26
1981	84

y = `35 + 0.4/(1!) (7) + ((0.4)(0.4 - 1))/(2!) (9) + (0.4(0.4 - 1)(0.4 - 2))/(3!)`

= `35 + 2.8/1 + ((0.4)(- 0.6)(9))/2 + ((0.4)(- 0.6)(- 1.6))/6`

y = 35 + 2.8 – 1.08 + 0.064

= 37.864 – 1.08

y = 36.784

∴ Population in the year 1955 is 36.784 (lakhs)