Question

If $a,b,c$ be the $p^{\text{th }},q^{\text{th }},r^{\text{th }}$ terms of a GP then the value of $(q-r)\log a+(r-p)\log b+(p-q)\log c$ is : (a) 0 (b) 1 (c) $-1$ (d) $pqr$

Short answer

Answer: (a) 0

Step-by-step solution

Find the formula for the nth term of a GP

We know that for a GP, the nth term is given by $T_n=a{r}^{(n-1)}$, where $a$ is the first term, $r$ is the common ratio, and $n$ is the term number.

Write expressions for $a,b,c$

Using the formula for the nth term of a GP, we have the following expressions: - $a=a{r}^{(p-1)}$ - $b=a{r}^{(q-1)}$ - $c=a{r}^{(r-1)}$

Rewrite the given expression using $a,b,c$

Now, we need to find the value of $(q-r)\log a+(r-p)\log b+(p-q)\log c$. We can rewrite this expression as: $(q-r)\log (a{r}^{(p-1)})+(r-p)\log (a{r}^{(q-1)})+(p-q)\log (a{r}^{(r-1)})$

Distribute the logs and simplify

Using the properties of logarithms, we can distribute the logs as follows: $$((q-r)(p-1)\log r+(r-p)(q-1)\log r+(p-q)(r-1)\log r$$ Now, we can factor out $\log r$: $$\log r[(q-r)(p-1)+(r-p)(q-1)+(p-q)(r-1)]$$ Next, expand the terms in the parenthesis: $$\log r(pq-pr-rq+rp+qr-pq+qr-rp+pq-rq)$$ Now, simplify by canceling out terms: $$\log r(pq-pr-rq+rp+qr-pq+qr-rp+pq-rq)=0$$ Since the value of the expression is 0, the correct choice is: (a) 0

Key concepts

Logarithmic Expressions

Logarithmic expressions are mathematical expressions involving logarithms. They help to transform multiplicative relationships into additive ones, making complex calculations easier. A logarithm, represented as ${\log }_b(x)$, is the exponent to which the base $b$ must be raised to produce the number $x$.
For instance, ${\log }_2(8)=3$ because $2^3=8$. Logarithms have several key properties which are useful for simplification and calculation:

• Product Rule: ${\log }_b(xy)={\log }_b(x)+{\log }_b(y)$
• Quotient Rule: ${\log }_b\left(\frac{x}{y}\right)={\log }_b(x)-{\log }_b(y)$
• Power Rule: ${\log }_b(x^y)=y\cdot {\log }_b(x)$

These properties are instrumental in breaking down and simplifying logarithmic expressions in problems similar to the one in our exercise. By applying these rules, expressions can be reformulated to more manageable forms, allowing for easier solution finding.

nth Term of a Sequence

In a geometric progression (GP), the nth term refers to the term at a specific position $n$ in the sequence. The formula for the nth term is $T_n=a{r}^{(n-1)}$, where:

• $a$ is the first term in the sequence,
• $r$ is the common ratio,
• $n$ is the term number.

Understanding the nth term's formula is crucial as it allows us to describe any term in the progression using its position number $n$.
In the exercise, $a,b,$ and $c$ are identified as specific terms in the GP, each corresponding to different positions $p,q,$ and $r$.
Thus:

• $a=a{r}^{(p-1)}$ for the $p^{\text{th}}$ term,
• $b=a{r}^{(q-1)}$ for the $q^{\text{th}}$ term,
• $c=a{r}^{(r-1)}$ for the $r^{\text{th}}$ term.

This formulation aids in linking each term of a GP with logarithmic operations, as demonstrated in the original solution.

Simplification Techniques

Simplification techniques are essential for breaking down complex expressions into simpler, solvable parts. Here, simplification is achieved by employing the fundamental properties of logarithms. These techniques allow us to concentrate on the core problem, removing extraneous complexity.
In the problem solution, we use:

• The Distributive Property of multiplication over addition: separating $\log (a{r}^{(n-1)})$ into $\log (a)+(n-1)\log (r)$.
• The ability to Factor common elements, like $\log r$, simplifying the aggregation of terms in parentheses.

Through simplification, terms with canceling effects become evident, leading to a zero-valued expression. This aligns with our ability to transform and simplify logarithmic expressions and confirms the correctness of the chosen answer in the exercise.