Question

If \[\dfrac{4}{5}\], a, 2 three consecutive term of an A.P then find ‘a’

Answer 1

Hint: Here in this question, we have a sequence that belongs to an arithmetic sequence. We check the sequence with the help of arithmetic sequence definition and it is an arithmetic sequence we can determine the common difference of the sequence. then we find the value of a.

Complete step-by-step solution:
In the sequence we have three kinds of sequence namely, arithmetic sequence, geometric sequence and harmonic sequence. 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.

The general arithmetic progression is of the form \[a,a + d,a + 2d,...\] where a is first term nth d is the common difference. The nth term of the arithmetic progression is defined as \[{a_n} = {a_0} + (n - 1)d\]
Now let us consider the sequence which is given in the question \[\dfrac{4}{5}\], a, 2 here we have 3 terms. Let us find the difference between these two consecutive numbers.
Here \[{a_1} = \dfrac{4}{5}\], \[{a_2} = a\], \[{a_3} = 2\]
Let we find
The difference between \[{a_1}\]and \[{a_2}\], so we have
\[d = {a_2} - {a_1} = a - \dfrac{4}{5}\] ------- (1)
The difference between \[{a_2}\]and \[{a_3}\], so we have
\[d = {a_3} - {a_2} = 2 - a\] ----- (2)
Let we equate the both equation (1) and the equation (2)
 \[ \Rightarrow a - \dfrac{4}{5} = 2 - a\]
Group the terms which are containing variable a to LHS and the constants to RHS.
\[ \Rightarrow a + a = 2 - \dfrac{4}{5}\]
On simplifying we have
\[ \Rightarrow 2a = \dfrac{{10 - 4}}{5}\]
\[ \Rightarrow 2a = \dfrac{6}{5}\]
Dividing the above equation by 2 we get
\[ \Rightarrow a = \dfrac{2}{5}\]
Hence we have got the value of a for the consecutive numbers.

Therefore the arithmetic sequence is \[\dfrac{4}{5},\dfrac{2}{5},2\]

Note: By considering the formula of arithmetic sequence we verify the obtained value 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.