Question

Which of the following form a sequence?
[a] 4,11,18,25,…
[b] 43,32,21,10,…
[c] 27,19,40,70,…
[d] 7,21,63,189,…

Answer 1

- Hint: Recall the definition of a sequence. An enumerated list of numbers in which repeated letters are allowed, and order matters is a sequence. Check which of the given lists satisfy the definition of a sequence.

Complete step-by-step solution -

We know that any enumerated list of numbers in any order in which repetition is allowed, and order matters is called a sequence. Some of the well-studied sequences in mathematics are Arithmetic sequence, Geometric sequence, Harmonic sequence, Fibonacci sequence. It is not necessary that the sequence has a well defined general term.
Keeping the definition in mind, we have,
4,11,18,25, … is an enumerated list of numbers and order is material.
Also, 11-4 = 18 -11 = 25 -18.
So, 4,11,18,25, … is a sequence and the adjacent terms differ by a fixed constant. Such a sequence is called an arithmetic sequence.
Similarly 43,32,21,10, … is a sequence, and since the adjacent terms differ by a fixed constant, the given sequence is an arithmetic sequence.
27,19, 40, 70 ,… is an enumerated list of numbers and the order in which numbers occur is material. Hence 27,18,40,70,… is also a sequence.
7,21, 63, 189 is also a sequence. Observe that the ratio of adjacent terms is a fixed constant. Such a sequence is called a geometric sequence.

Hence all of the options [a], [b] , [c] and [d] are correct.

Note: There is a common misconception between sequences and series.
A sequence is an enumerated list of numbers in which order matters, whereas a series is the sum of the terms of a sequence.
Hence 4,11,18,25,… is a sequence, whereas 4 +11 +18 +25 +… is a series.