Chapter 4

Series: Binomial, Arithmetic and Geometric (Pure 1)

The binomial expansion for a positive integer power, arithmetic and geometric progressions, and the condition that decides whether an infinite sum exists at all.

หลักสูตรแกนกลาง / สสวท.: จำนวนและพีชคณิต (Number and Algebra)ตรงกับ ม.5 ลำดับและอนุกรม (เพิ่มเติม)
Common Core (US): A-SSE, A-APR, F-BF
Cambridge: 9709 Pure Mathematics 1 — Series

4.1 The binomial expansion

For a positive integer $n$

(a + b)^{n} = r = 0 \sum n^{n} C_{r} a^{n - r} b^{r} T_{r + 1} =^{n} C_{r} a^{n - r} b^{r}

The expansion stops. That is the whole difference between this and the Pure 3 version, where the power is not a positive integer and the series runs forever.

ตัวอย่าง

Find the coefficient of $x^{3}$ in the expansion of $(2 - 3 x)^{6}$ .

วิธีทำ

1.Term: $^{6} C_{3} (2)^{3} (- 3 x)^{3}$
2. $^{6} C_{3} = 20$
3. $20 \times 8 \times (- 27)$

ตอบ $- 4320$

ข้อควรระวัง

$(- 3 x)^{3}$ is $- 27 x^{3}$ , not $- 3 x^{3}$ . The bracket carries the coefficient into the power along with the sign, and losing it is the standard way this question goes wrong.

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