Question

In a GP, if the fourth term is the square of the second term, then the relation between the first term and common ratio is (1) \(\mathrm{a}=\mathrm{r}\) (2) \(\mathrm{a}=2 \mathrm{r}\) (3) \(2 \mathrm{a}=\mathrm{r}\) (4) \(\mathrm{r}^{2}=\mathrm{a}\)

Short answer

Question: In a geometric progression, the fourth term is the square of the second term. Find the relation between the first term (a) and the common ratio (r). Answer: The relation between the first term and the common ratio is a = r.

Step-by-step solution

Find the Formula for Second and Fourth Term

In the given GP, let's denote the first term as "a" and the common ratio as "r". The formula for the nth term of a GP is: $$T_n = ar^{n-1}$$ For second term (n=2): $$T_2 = ar^1 = ar$$ And for the fourth term (n=4): $$T_4 = ar^{4-1} = ar^3$$

Use the Given Condition

The question states that the fourth term is the square of the second term. So, we can represent it mathematically as: $$T_4 = (T_2)^2$$ We have the formulas of \(T_2\) and \(T_4\) from the previous step, so we can substitute them: $$ar^3 = (ar)^2$$

Solve for the Relation Between a and r

Now we will solve the equation to find the relationship between a and r: $$ar^3 = a^2r^2$$ Divide both sides by \(ar^2\): $$\frac{ar^3}{ar^2} = \frac{a^2r^2}{ar^2}$$ Simplify to get the relation: $$r = a$$ So the relation between the first term and common ratio is (1) \(\mathrm{a}=\mathrm{r}\).

Key concepts

Common Ratio

The common ratio in a geometric progression (GP) is a fundamental concept that determines the relationship between consecutive terms in the sequence. For instance, in the sequence 2, 6, 18, 54, ..., each term after the first is obtained by multiplying the previous term by 3. Hence, the common ratio here is 3.

Understanding the common ratio can help students unravel the nature of a geometric sequence, whether it's increasing, decreasing, or constant. If the common ratio is greater than one, the sequence will increase; if it's between zero and one, the sequence will decrease; and if the ratio is negative, the sequence will alternate between positive and negative terms.

Nth Term of a GP

The nth term of a geometric progression provides a way to directly calculate the value of any term in the sequence without having to list all the preceding terms. It is denoted as T_n and is determined using the formula:
\[ T_n = ar^{n-1} \]
where a is the first term, r is the common ratio, and n is the term's position in the sequence.

This formula is instrumental when solving problems related to GPs since it encapsulates the essence of the progression in an algebraic expression. For example, knowing the first term and the common ratio is enough to find any term in the sequence without iterative multiplication.

Sequences and Series

Sequences and series are foundational concepts in mathematics, especially when studying patterns and their applications. A sequence is an ordered list of numbers generated by a specific rule, such as a geometric progression generated by repeatedly multiplying by the common ratio.

A series, on the other hand, is the sum of the terms of a sequence. In the case of a geometric series, it is the sum of the terms of a geometric progression. An understanding of sequences and series is not only crucial for solving mathematical problems but also has practical applications in fields like finance, computer science, and physics.

Geometric Series Formula

For a finite geometric series, the sum can be calculated using the formula:
\[ S_n = \frac{a(1 - r^n)}{1 - r} \]
where S_n is the sum of the first n terms, a is the first term, r is the common ratio, and n is the number of terms to be added.