The first term of an Arithmetic Progression is 20. If the fifth term is four-fifth of the third term, find the common difference.

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Question 5: The first term of an Arithmetic Progression is 20. If the fifth term is four-fifth of the third term, find the common difference.

Step 1: Write the terms of the Arithmetic Progression (AP).
Let the first term be $a$ and the common difference be $d$.
Given $a = 20$.
The third term is $a_3 = a + (3-1)d = a + 2d = 20 + 2d$.
The fifth term is $a_5 = a + (5-1)d = a + 4d = 20 + 4d$.

Step 2: Use the given condition to form an equation.
Given that the fifth term is four-fifth of the third term:

$$a_5 = \frac{4}{5} a_3$$ $$20 + 4d = \frac{4}{5}(20 + 2d)$$

Step 3: Solve the equation for $d$.
Multiply both sides by 5:

$$5(20 + 4d) = 4(20 + 2d)$$ $$100 + 20d = 80 + 8d$$

Subtract $8d$ from both sides:

$$100 + 12d = 80$$

Subtract 100 from both sides:

$$12d = 80 - 100$$ $$12d = -20$$ $$d = \frac{-20}{12}$$ $$d = -\frac{5}{3}$$

The correct option is A.
The final answer is $\boxed{\text{A. } -\frac{5}{3}}$.

Question 6: Evaluate $2\log 4 + 3\log 5 - \log 2$.

Step 1: Apply the logarithm power rule $n \log a = \log a^n$.

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