Construct a forward difference table for the following data

x	0	10	20	30
y	0	0.174	0.347	0.518

Construct a forward difference table for y = f(x) = x³+2x+1 for x = 1,2,3,4,5

By constructing a difference table and using the second order differences as constant, find the sixth term of the series 8,12,19,29,42…

Find (i) Δe^ax
(ii) Δ²e^x
(iii) Δ log x

Evaluate \(\Delta \)\(\left[ \frac { 5x+12 }{ { x }^{ 2 }+5x+6 } \right] \) by taking ‘1’ as the interval of differencing.

Prove that f(4) = f(3) + Δf(2) + Δ² f(1) + Δ³ f(1) taking ‘1’ as the interval of differencing.

Given U₀ = 1, U₁ = 11, U₂ = 21, U₃ = 28 and U₄ = 29 find Δ⁴U₀

Given y₃ = 2, y₄ = −6, y₅ = 8, y₆ = 9 and y₇ = 17 Calculate Δ⁴y₃

From the following table find the missing value

x	2	3	4	5	6
f(x)	45.0	49.2	54.1	-	67.4

Estimate the production for 1964 and 1966 from the following data

Year	1961	1962	1963	1964	1965	1966	1967
Production	200	220	260	-	350	-	430