Question

If \(\mathrm{a}, \mathrm{b}, \mathrm{c}\) are in AP as well as in GP, then (a) \(\mathrm{a}=\mathrm{b} \neq \mathrm{c}\) (b) \(\mathrm{a} \neq \mathrm{b}=\mathrm{c}\) (c) \(\mathrm{a} \neq \mathrm{b} \neq \mathrm{c}\) (d) \(\mathrm{a}=\mathrm{b}=\mathrm{c}\)

Short answer

Answer: (a) a = b ≠ c and (d) a = b = c

Step-by-step solution

Arithmetic Progression Condition

If a sequence is in arithmetic progression, the common difference between consecutive terms is constant for any consecutive terms. In this case, the common difference (d) is the same for terms a and b, as well as for terms b and c. Therefore, we can write: b - a = c - b

Geometric Progression Condition

If a sequence is in geometric progression, the common ratio between consecutive terms remains constant for any consecutive terms. In this case, the common ratio (r) for terms a and b is the same as for terms b and c. Therefore, we can write: b/a = c/b Now, we can analyze each answer choice:

Option (a)

a = b ≠ c AP condition: b - a = c - b => 0 = c - b => c = b GP condition: b/a = c/b => b^2 = a*c => b^2 = (b)(b) This condition is satisfied as a = b and a ≠ c.

Option (b)

a ≠ b = c AP condition: b - a = c - b => b - a = 0 => a = b which contradicts this condition, hence option (b) is not correct.

Option (c)

a ≠ b ≠ c AP condition: b - a = c - b GP condition: b/a = c/b => b^2 = a*c => b^2 ≠ a*a and b^2 ≠ c*c (since a ≠ b ≠ c) Since b^2 ≠ a*a and b^2 ≠ c*c, we cannot satisfy the conditions for both AP and GP. Therefore, option (c) is not correct.

Option (d)

a = b = c AP condition: b - a = c - b => 0 = 0 GP condition: b/a = c/b => 1 = 1 This condition is satisfied as all terms are equal. Based on our analysis, we conclude that the correct answer is #(a) a = b ≠ c and (d) a = b = c#.

Key concepts

arithmetic progression

An arithmetic progression (AP) is a sequence of numbers in which the difference between consecutive terms is constant. This difference is known as the common difference. Understanding AP is crucial as it represents linear growth or decay. Imagine having a series like 2, 4, 6, 8; here, the gap between each term is 2, which is our common difference. This consistent increment characterizes the arithmetic sequence.

Key Points about Arithmetic Progression:

• The formula for the n-th term is given by: \[ a_n = a_1 + (n-1) \cdot d \] where \( a_1 \) is the first term and \( d \) is the common difference.
• The sum of the first n terms is called an arithmetic series and is calculated as: \[ S_n = \frac{n}{2} \cdot (2a_1 + (n-1)d) \]
• Arithmetic progressions are linear patterns, vital in various real-life calculations, such as loan payments or savings.

geometric progression

A geometric progression (GP) is a sequence where each term after the first is obtained by multiplying the previous one by a non-zero common ratio. This type of sequence highlights exponential growth or decay, playing a major role in fields like population studies or compound interest.

Characteristics of Geometric Progression:

• Mathematically, the n-th term of a GP is expressed as: \[ a_n = a_1 \cdot r^{(n-1)} \] where \( a_1 \) is the first term and \( r \) is the common ratio.
• The sum of the first n terms is given by: \[ S_n = a_1 \cdot \frac{1-r^n}{1-r} \] if \( r eq 1 \).
• In our exercise, if terms are both in AP and GP, it typically leads to the terms being consistent or repeating, like the option \( a = b = c \).

common difference

The common difference is a cornerstone concept in arithmetic progressions. It’s the fixed amount added to each term to get the next. It defines the linear growth of a sequence and directly affects the nature and behavior of the sequence. Think of it as the backbone of an arithmetic procession’s structure.

Insights into Common Difference:

• A positive common difference implies an increasing sequence, while a negative one results in a decreasing sequence.
• If the common difference is zero, the sequence becomes constant, with all terms equal.
• Understanding the common difference helps in predicting future terms without listing each one, which is useful in quick calculations or identifying patterns.