3.1. Find the first 3 terms of the Binomial expansion of .
3.2. If , where and are constants
and , find the value of and .

Ask by Schmidt Ramos. in South Africa

Mar 16,2025

Question

3.1. Find the first 3 terms of the Binomial expansion of .
3.2. If , where and are constants
and , find the value of and .

Ask by Schmidt Ramos. in South Africa

Mar 16,2025

UpStudy AI · Accepted Answer

**Step 1. Write the binomial expansion**

We start with the binomial expansion formula:
$$
(a+b)^n=\sum_{r=0}^{n}\binom{n}{r}a^{n-r}b^r.
$$
For $$(2-kx)^8$$, let $$a=2$$ and $$b=(-kx)$$, so the expansion is:
$$
(2-kx)^8 = \sum_{r=0}^{8}\binom{8}{r}2^{8-r}(-kx)^r.
$$

**Step 2. Find the first three terms**

- **Term with $$r=0$$:**
  $$
  \binom{8}{0}2^8(-kx)^0 = 1\cdot256\cdot1 = 256.
  $$

- **Term with $$r=1$$:**
  $$
  \binom{8}{1}2^7(-kx) = 8\cdot128\cdot(-kx) = -1024kx.
  $$

- **Term with $$r=2$$:**
  $$
  \binom{8}{2}2^6(-kx)^2 = 28\cdot64\cdot k^2x^2.
  $$
  Since $$(-kx)^2=k^2x^2$$, the term simplifies to:
  $$
  28\cdot64\cdot k^2x^2 = 1792k^2x^2.
  $$

Thus, the first three terms of the expansion are:
$$
256 - 1024kx + 1792k^2x^2.
$$

**Step 3. Compare with the given expansion**

We are given:
$$
(2-kx)^8 = 256 + Ax + 1008x^2 + \cdots,
$$
so by comparing coefficients:

- **Coefficient of $$x$$:**
  $$
  A = -1024k.
  $$

- **Coefficient of $$x^2$$:**
  $$
  1792k^2 = 1008.
  $$

**Step 4. Solve for $$k$$**

Solve the equation:
$$
1792k^2 = 1008.
$$
Divide both sides by 1792:
$$
k^2 = \frac{1008}{1792}.
$$
Simplify the fraction by dividing numerator and denominator by 16:
$$
k^2 = \frac{1008 \div 16}{1792 \div 16} = \frac{63}{112}.
$$
Further simplify by dividing numerator and denominator by 7:
$$
k^2 = \frac{63 \div 7}{112 \div 7} = \frac{9}{16}.
$$
Since $$k>0$$,
$$
k = \sqrt{\frac{9}{16}} = \frac{3}{4}.
$$

**Step 5. Find $$A$$**

Now substitute $$k=\frac{3}{4}$$ into the expression for $$A$$:
$$
A = -1024k = -1024\left(\frac{3}{4}\right) = -768.
$$

**Final Answers:**

3.1. The first three terms of the expansion are:
$$
256 - 1024kx + 1792k^2x^2.
$$

3.2. The constant values are:
$$
k = \frac{3}{4} \quad \text{and} \quad A = -768.
$$
The first three terms of the expansion are $$256 - 1024kx + 1792k^2x^2$$. The values are $$k = \frac{3}{4}$$ and $$A = -768$$.

3.1. Find the first 3 terms of the Binomial expansion of .
3.2. If , where and are constants
and , find the value of and .

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