RI Sec 3 EOY 1998 Q8
Write down and simplify the first three terms in the expansion of (1 - 2x)8 in ascending powers of x. Given that the first three terms in the expansion of (1 - ax)2 (1 - 2x)8 are 1, -8x, and bx2, calculate the values of a and b.
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Answer:
(1 - 2x)8
= 8C0 (1)8 (-2x)0 + 8C1 (1)7 (-2x)1 + 8C2 (1)6 (-2x)2 + ...
= 1 - 16x + 112x² + ...
(1 - ax)2 (1 - 2x)8
= (1 - 2ax + a²x²) (1 - 16x + 112x² + ...)
= 1 - 2ax + a²x² - 16x + 32ax² + 112x² + ...
= 1 - (2a + 16)x + (a² + 32a + 112)x² + ...
Comparing coefficients,
-(2a + 16) = -8
2a + 16 = 8
2a = -8
a = -4
b = a² + 32a + 112
b = 16 - 128 + 112
b = 0
Thus, a = -4, b = 0
