Question

Using interpolation estimate the business done in 1985 from the following data

Year	1982	1983	1984	1986
Business done (in lakhs)	150	235	365	525

Answer 1

Here the intervals are unequal.

By Lagrange’s formula we have,

x₀ = 1982

x₁ = 1983

x₂ = 1984

x₃ = 1986

y₀ = 150

y₁ = 235

y₂ = 365

y₃ = 525 and x = 1985.

y = `"f"(x) = ((x - x_1)(x - x_2)(x - x_3))/((x_0 - x_1)(x_0 - x_2)(x_0 - x_3)) xx y_0 + ((x - x_0)(x - x_2)(x - x_3))/((x_1 - x_0)(x_1 - x_2)(x_1 - x_3)) xx y_1 + ((x - x_0)(x - x_1)(x - x_3))/((x _2 - x_0)(x_2 - x_1)(x_2 - x_3)) xx y_2 + ((x - x_0)(x - x_1)(x - x_2))/((x_3 - x_0)(x_3 - x_1)(x_3 - x_2)) xx y_3`

y = `((1985 - 1983)(1985 - 1984)(1985 - 1986))/((1982 - 1983)(1982 - 1984)(1982 - 1986)) xx 150 + ((1985 - 1982)(1985 - 1984)(1985 - 1986))/((1983 - 1982)(1983 - 1984)(1983 - 1986)) xx 235  ((198  - 982)(1985 - 1983)(985 - 1986))/((984 - 1982)(1984 - 1983)(1984 - 1986)) xx 365 + ((1985 - 1982)(1985 - 1983)(1985 - 1984))/((1986 - 1982)(1986 - 1983)(1986 - 1984)) xx 525`

= `((2)(1)(-1))/((-1)(-2)(-4)) xx 150 + ((3)(1)(-1))/((1)(-1)(-3)) xx 235 + ((3)(2)(-1))/((2)(1)(-)) xx 365  + ((3)(2)(1))/((4)(3)(2)) xx 525`

= `(-2 xx 150)/(-8) + (-3 xx 235)/3 + (-6 xx 365)/(-4) xx (6 xx 525)/24`

= 37.5 – 235 + 547.5 + 131.25

= 716.125 – 235

= 481.25

∴ Business done in the year 1985 is 481.25 lakhs.