If α = 
and f(x) = A 0 + 
, then find the value of, f(x) + f(αx) + ..... + f(α 6 x) independent of α.
Text Solution
Verified by ExpertsCHECK THE SOLUTION.
Sol. f(x) = A 0 + 
= A 0 + A 1 .x + A 2 x 2 + ...... + A 20 x 20
f(αx) = A 0 + A 1 .α.x + A 2 .α 2 .x 2 + ...... + A 20 .α 20 .x 20
f(α 2 x) = A 0 + A 1 .α 2 .x + A 2 .α 4 .x 2 + ...... + A 20 .α 40 .x 20
f(α 6 x) = A 0 + A 1 .α 6 .x + A 2 .α 12 .x 2 + ...... + A 20 .α 120 . x 20 f(x) + f(αx) + f(α
2 x) + .... + f(α 6 x)
= 7A 0 + A 1 x (1 + α + α 2 + .... + α 6 ) + A 2 x 2 (1 + α 2 + α 4 + .... + α 12 ) + ... + A 20 x 20 (1 + α 20 + α 40 + ... + α 120 )
We know
1 + α p + α 2p + ... + α 6p = 
∴ sum = 7A 0 + 7A 7 x 7 + 7A 14 x 14
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