Question 5 Appropriate formulae must be used in this question. If $$ \frac{a+b}{a-b}=\frac{b+c}{b-c} $$ show that $$ a, b $$ and $$ c $$ form a geometric sequence

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UpStudy AI · Accepted Answer

To show that $$ a $$, $$ b $$, and $$ c $$ form a geometric sequence given the equation:

$$
\frac{a + b}{a - b} = \frac{b + c}{b - c}
$$

we can follow these steps:

### Step 1: Cross-Multiply the Equation

Start by cross-multiplying to eliminate the fractions:

$$
(a + b)(b - c) = (a - b)(b + c)
$$

### Step 2: Expand Both Sides

Expand both sides of the equation:

$$
a(b - c) + b(b - c) = a(b + c) - b(b + c)
$$

$$
ab - ac + b^2 - bc = ab + ac - b^2 - bc
$$

### Step 3: Simplify the Equation

Subtract $$ ab $$ from both sides:

$$
-ab - ac + b^2 - bc = ab + ac - b^2 - bc - ab
$$

Simplify further:

$$
-ab - ac + b^2 - bc = ac - b^2 - bc
$$

Add $$ ac $$ and $$ b^2 $$ to both sides:

$$
b^2 = ac
$$

### Step 4: Interpret the Result

The equation $$ b^2 = ac $$ is the defining condition for a geometric sequence. In a geometric sequence, the square of the middle term is equal to the product of the terms on either side. Therefore, $$ a $$, $$ b $$, and $$ c $$ form a geometric sequence.

### Conclusion

Given the initial condition $$ \frac{a + b}{a - b} = \frac{b + c}{b - c} $$, we have successfully shown through algebraic manipulation that $$ a $$, $$ b $$, and $$ c $$ satisfy the condition $$ b^2 = ac $$, confirming that they form a geometric sequence.

**Answer:**

After cross-multiplying and simplifying, we find that b² = a c. Thus, a, b, c satisfy b² = a c and form a geometric sequence.
$$ a $$, $$ b $$, and $$ c $$ form a geometric sequence because $$ b^2 = a \times c $$.

Question 5 Appropriate formulae must be used in this question. If \( \frac{a+b}{a-b}=\frac{b+c}{b-c} \) show that \( a, b \) and \( c \) form a geometric sequence

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