Cost of living indices of a Banglore for some years are given below. I

1.	Cost of living indices of a Banglore for some years are given below. Interpolate the missing index number for and
Answer» Since the known values are 5, the fifth leading differences will be zero, i.e. Δ⁵ = 0 Δ⁵ = (y – 1 )⁵ = y₅ – 5y₄ +10y₃ – 10y₂ + 5y₁ – y₀ = 0 …….(i) And the second equation can be obtained by, increasing the suffixes of each term of’y’ by one, keeping the coefficients same; ie. (y- 1)⁵ = y₆ – 5y₅ + 10y₄ – 10y₃+ 5y₂ – y, = 0 ………. (ii) determine the value of Y₂ from equation (i) 162 – 5(142) +10 (128) – 10y₂+ 5(112) – 100 = 0 by simplifying, -10y₂ = -1192 y₂ = 119.2 Hence, the missing Index number for 1990 is 119.2 From (ii) y₆ – 5(162) + 10(142) – 10(128) + 5(119.2) – 112 = 0 Here y₂ =119.2 y₆ – 810 + 1420 – 1280 + 596- 112 = 0. ∴ y₆ = 186 Hence the cost of living index number for the year 2010 is 186.

Cost of living indices of a Banglore for some years are given below. Interpolate the missing index number for and

Answer»

Since the known values are 5, the fifth leading differences will be

zero, i.e. Δ⁵ = 0

Δ⁵ = (y – 1 )⁵

= y₅ – 5y₄ +10y₃ – 10y₂ + 5y₁ – y₀ = 0 …….(i)

And the second equation can be obtained by, increasing the suffixes of each term of’y’ by one, keeping the coefficients same;

ie. (y- 1)⁵ = y₆ – 5y₅ + 10y₄ – 10y₃+ 5y₂ – y, = 0 ………. (ii)

determine the value of Y₂ from equation (i)

162 – 5(142) +10 (128) – 10y₂+ 5(112) – 100 = 0

by simplifying, -10y₂ = -1192

y₂ = 119.2

Hence, the missing Index number for 1990 is 119.2

From (ii) y₆ – 5(162) + 10(142) – 10(128) + 5(119.2) – 112 = 0

Here y₂ =119.2

y₆ – 810 + 1420 – 1280 + 596- 112 = 0.

∴ y₆ = 186

Hence the cost of living index number for the year 2010 is 186.