Question

How do you identify \[{a_3}\] of this sequence: 0.25, 0.5, 0.75, 1, 1.25, 1.5?

Answer 1

Hint: Here in this question, we have a sequence and we have to check whether the sequence belongs to an arithmetic sequence or not. We check the sequence with the help of arithmetic sequence definition and if it is an arithmetic sequence we determine the common difference of the sequence. Then we determine the third term of the sequence.

Complete step-by-step solution:
i) In the sequence we have three kinds of sequence namely, arithmetic sequence, geometric sequence and harmonic sequence.
ii) In arithmetic sequence we the common difference between the two terms, In geometric sequence we the common ratio between the two terms, In harmonic sequence it is a ratio of arithmetic sequence to geometric sequence.
iii) The general arithmetic progression is of the form \[a,a + d,a + 2d,...\] where a is first term and d is the common difference. The nth term of the arithmetic progression is defined as \[{a_n} = {a_0} + (n - 1)d\]
The given set of numbers should be any one of the sequences then we determine the third element \[{a_3}\].

Now let us consider the sequence which is given in the question 0.25, 0.5, 0.75, 1, 1.25, 1.5 where we have 6 terms. Let us find the difference between these two consecutive numbers.
Let we find
The difference between first and second term, so we have
\[0.5 - 0.25 = 0.25\]
The difference between second and third term, so we have
\[0.75 - 0.5 = 0.25\]
The difference between third and fourth term, so we have
\[1 - 0.75 = 0.25\]
The difference between fourth and fifth term, so we have
\[1.25 - 1 = 0.25\]
The difference between fifth and sixth term, so we have
\[1.5 - 1.25 = 0.25\]
Hence we have got the same difference for the consecutive numbers.
Therefore the given sequence is an arithmetic sequence
The common difference of the arithmetic sequence 4, 7, 10, 13, 16,… is 0.25
The third element \[{a_3}\] of the above sequence is 0.75.


Note: By considering the formula of arithmetic sequence we verify the common difference which we obtained. We have to check the common difference for all the terms. Suppose if we check for the first two terms not for other terms then we may go wrong. So definition of arithmetic sequence is important to solve these kinds of problems. The sequence must be ordered or arranged in a particular way.