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354, Numerical Analysis, RHS =, %3D, I+x (1+x)?, (1 +x)³, 1+x (1+x)² (1÷r, 1, 1+x, A, 1+x(I÷A)), 1+, 1+x, = (1+xE)Auo = LHS, LHS = RHS, 1+xE, Example 4: Prove that uo +u¡x÷Uzr²÷... to =, (1-x) (1-r)?, .to%3D, (1-x}, Solution : uo+u,x+U2x²+. . . to o = ug+xEuo÷xE*uo÷. .. to, 1, = (1+xE+x*E²+... to o)uo, -1+XE÷X*E²÷... are in G.P. S, I- XE, 1, 1, 0n-, 1- x(1+A), 1-, (1-x), %3D, 1-x- xA, (1-x) 1-, 1-., I, 1+, 1-x, to uo, (1-x)?, Auo, (1-x)3, ...to e = RHS, %3D, 1-x (1-x), Note : The above result is known as Montmorts theorem., Example 5: Use the method of separation of symbols prove, -1,, uz =Uy-1+Aux-2+...+A"-ug-n +A®u-n, Solution : RHS uz-1+Aux-2+...+A"-u-n +4°u-n, E-lu̟ +AE²u,+.. .+Aª-'E™®u, +A®E¯®u,, 4n-1), 1+, t., +., E E?, = E, E", A" -E", En-I, 1, U;ミ, E", un, ER, = E, A-E, Scanned with CamScanner

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356, Numerical Analysis, 4(1+A)+A, (2+4)2, 4E, %3D, %3D, 4E, 4E, (2+4)², - ZA, 2E2 1, = 2E2., = E, Ur, (2+A), ", 1-, +-4" u, = RHS, r+-, 2, EXERCISE, By using the method of separation of symbols, prove the following :, 1, 1, 1, 1) uo-u + u2-u3+...=, Auo +A²uo-APuo+..., 4, 16, U, - "C¡ U,x-1 + "C2ur-2-"C3ur-3+... = A"ux-n, 3) un -"C, 2u2n-1+ "C22²u2n-2-...+(-2)"u, = (-1)" (c– 2an), 2), If u= ar-+br+c, 1x3(, 3, 2! (8, 4) ug =u+i +Ux÷2-Ur+3+, 3.7 interpolation :, Introduction : Interpolation is the technique of estimating the value of a function for any, intermediate value of the independent variable. It is the art of reading between the lines of a, table., For example we are given the foillowing data, The, of finding f(1956), f(1964), f(1975), рrоcess, etc.,' is known assinterpolation., 1941, 1951, 1961, 1971, 1981, f(r), 12, 15, 20, 27, 39, Interpolation technique is used in various disciplines like Economics, business, populaion, studies, the price determination etc. It is used to fill in the gaps in the statistical data for the, sake of continuity of information. For example if we know the records for the past five yerrs, except the third year which is not available due to unforeseen conditions, the interpoi:ation, technique helps to estimate the record for that year too under the assumption that the changes in, the records over these five years have been uniform,, Scanned with CamScanner