Identifying Sequences: Which One is Not an Arithmetic Progression?

An Arithmetic Progression (A.P.) is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference, usually denoted by \(d\).

To determine which of the given sequences is not an A.P., we need to calculate the difference between consecutive terms for each sequence. If the difference remains constant throughout the sequence, it is an A.P.; otherwise, it is not.

Analyzing Each Sequence for Arithmetic Progression

Sequence 1: 11, 7, 3, -1, -5

• Difference between 2nd and 1st term: \(7 - 11 = -4\)
• Difference between 3rd and 2nd term: \(3 - 7 = -4\)
• Difference between 4th and 3rd term: \(-1 - 3 = -4\)
• Difference between 5th and 4th term: \(-5 - (-1) = -5 + 1 = -4\)

The difference between consecutive terms is consistently \(-4\). Thus, this sequence is an A.P. with a common difference \(d = -4\).

Sequence 2: 0, -3, -6, -9

• Difference between 2nd and 1st term: \(-3 - 0 = -3\)
• Difference between 3rd and 2nd term: \(-6 - (-3) = -6 + 3 = -3\)
• Difference between 4th and 3rd term: \(-9 - (-6) = -9 + 6 = -3\)

The difference between consecutive terms is consistently \(-3\). Thus, this sequence is an A.P. with a common difference \(d = -3\).

Sequence 3: 3, 9, 19, 33, 51

• Difference between 2nd and 1st term: \(9 - 3 = 6\)
• Difference between 3rd and 2nd term: \(19 - 9 = 10\)
• Difference between 4th and 3rd term: \(33 - 19 = 14\)
• Difference between 5th and 4th term: \(51 - 33 = 18\)

The differences between consecutive terms are \(6, 10, 14, 18\). These differences are not constant. Therefore, this sequence is not an A.P.

Sequence 4: 9, 13, 17, 21, 25

• Difference between 2nd and 1st term: \(13 - 9 = 4\)
• Difference between 3rd and 2nd term: \(17 - 13 = 4\)
• Difference between 4th and 3rd term: \(21 - 17 = 4\)
• Difference between 5th and 4th term: \(25 - 21 = 4\)

The difference between consecutive terms is consistently \(4\). Thus, this sequence is an A.P. with a common difference \(d = 4\).

Conclusion on Identifying the Non-Arithmetic Progression

Based on our analysis, three of the sequences have a constant common difference, fulfilling the definition of an Arithmetic Progression. The sequence 3, 9, 19, 33, 51 shows varying differences between consecutive terms.

Therefore, the sequence that is not an A.P. is 3, 9, 19, 33, 51.

Revision Table: Key Concepts of Arithmetic Progression

Additional Information: Understanding Sequences and Series

A sequence is an ordered list of numbers. Each number in the sequence is called a term. Sequences can follow various patterns, such as arithmetic progressions, geometric progressions, or others.

An Arithmetic Progression is a specific type of sequence where the pattern is adding a constant value (the common difference) to get the next term. The common difference can be positive (for increasing A.P.), negative (for decreasing A.P.), or zero (for a constant sequence).

A series is the sum of the terms of a sequence. For example, the series corresponding to the A.P. 2, 4, 6 is 2 + 4 + 6 = 12. Understanding sequences and series is fundamental in many areas of mathematics.