Question

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in A. P, then \(7^{\mathrm{a}}, 7^{\mathrm{b},}, 7^{\mathrm{c}}\) are in (a) HP (b) AP (c) AGP (d) GP

Short answer

Question: The progression 7^a, 7^b, and 7^c is in ___________, if a, b, c are in arithmetic progression. Answer: Geometric Progression

Step-by-step solution

Understand the given information

We are given that a, b, and c are in an arithmetic progression. This means that b-a = c-b, or b = (a+c)/2.

Write given sequences

We are given the sequence 7^a, 7^b, and 7^c. Since b = (a+c)/2, we can rewrite the sequence as 7^a, 7^{(a+c)/2}, and 7^c.

Check for Harmonic Progression

A harmonic progression is a sequence whose reciprocals are in an arithmetic progression. We need to check the reciprocals of the given sequence: 1/7^a, 1/7^{(a+c)/2}, and 1/7^c. Since the reciprocals are in the form of exponents, they will not form an arithmetic progression. So, the sequence is not in harmonic progression.

Check for Arithmetic Progression

The given sequence is 7^a, 7^{(a+c)/2}, and 7^c. We need to check if the difference between consecutive terms forms a constant, i.e., if 7^{(a+c)/2} - 7^a = 7^c - 7^{(a+c)/2}. This condition is not satisfied for all values of a and c; therefore, the sequence is not an arithmetic progression.

Check for Geometric Progression

A geometric progression is a sequence in which the ratio between consecutive terms is constant. We need to check if the given sequence has a constant ratio, i.e., if (7^{(a+c)/2} / 7^a) = (7^c / 7^{(a+c)/2}). Using the properties of exponentiation, we can simplify the expression: (7^{(a+c)/2-a}) = (7^{c-(a+c)/2}), which simplifies to: (7^{(c-a)/2}) = (7^{(c-a)/2}). This condition is satisfied, so the sequence is a geometric progression.

Conclusion

Based on our analysis, we conclude that the sequence 7^a, 7^b, and 7^c is in a geometric progression. So, the correct answer is option (d) GP.

Key concepts

Geometric Progression

A geometric progression (GP) is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio. This simple rule can produce sequences that grow or shrink dramatically, which makes them quite interesting. For example, in the sequence 2, 4, 8, 16, each number is 2 times the number before it, making 2 the common ratio.

The fascinating part about geometric progressions arises when we use exponents. Notice that terms in a geometric sequence can also be represented using powers. This property is what we leveraged in our example problem where we needed to show that the sequence \(7^a, 7^b, 7^c\) is a geometric progression. By using the property of exponents that states \(x^m / x^n = x^{m-n}\), we were able to verify the constant ratio between terms. This ratio is vital in proving the sequence fits the definition of a geometric progression. When you understand this, many problems become much easier to solve.

Exponents

Exponents are a shorthand notation used to simplify expressions, especially those involving multiplication repeated that many times. For example, \(7^3\) means \(7 \times 7 \times 7\). Learning to work with exponents is crucial as they appear in many areas of mathematics, including sequences like the ones discussed in arithmetic and geometric progressions.

Using the rules of exponents efficiently can simplify the analysis of sequences. For instance, knowing that \(x^a \cdot x^b = x^{a+b}\) or \((x^a)^b = x^{a \cdot b}\) enables easy manipulation of the terms. In the context of the exercise, these rules helped us break down expressions like \(7^b\) where \(b = \frac{a+c}{2}\) into more manageable parts, proving that the sequence is indeed a geometric progression. The power of exponents lies not only in solving algebraic expressions but also in revealing patterns in sequences.

Sequence Analysis

Sequence analysis involves examining a list of numbers arranged in a specific order to decipher their properties and identify patterns. Sequences can follow various rules, such as arithmetic progressions (adding a constant), geometric progressions (multiplying by a constant), or even more complex relationships.

• Analyzing a sequence requires understanding its foundation. What rule does it follow?
• Does it have a common difference (AP), a common ratio (GP), or any other rule?

In our exercise, sequence analysis led us to classify the sequence \(7^a, 7^b, 7^c\) as a geometric progression. This required checking if the ratio between the terms was consistent. For students, practicing sequence analysis develops logical thinking and pattern identification skills, which are useful well beyond math classes. When dissecting sequences, remember: look for the pattern, apply the right concepts, and draw conclusions based on mathematical reasoning.