Question

Find the value of k such that \[2k-1,7,11\] are in arithmetic progression.

Answer 1

Hint: In this question, as given that the three terms are in arithmetic progression they tend to have the common difference which is given by the formula \[{{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{2}}\]. Now, on equating the common difference of the consecutive terms and simplifying further we get the result.

Complete step by step answer:
ARITHMETIC PROGRESSION (A.P)
A sequence in which the difference of two consecutive terms is constant, is called Arithmetic progression.
Where, a is called the first term of the series and d is called the common difference of the series.
Now, if we consider that there are three terms \[{{a}_{1}},{{a}_{2}},{{a}_{3}}\] which are said to be in arithmetic progression then we have the condition that
\[\Rightarrow {{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{2}}\]
Now, given terms that are in arithmetic progression from the question are \[2k-1,7,11\]
Now, on comparing those terms given in the question with the general terms we have
\[{{a}_{1}}=2k-1,{{a}_{2}}=7,{{a}_{3}}=11\]
Now, from the above mentioned condition we have
\[\Rightarrow {{a}_{2}}-{{a}_{1}}={{a}_{3}}-{{a}_{2}}\]
Now, on substituting the respective values in the above condition we get,
\[\Rightarrow 7-\left( 2k-1 \right)=11-7\]
Now, this can be further written in the simplified form as
\[\Rightarrow 7-2k+1=4\]
Now, on rearranging the terms on the both sides we get,
\[\Rightarrow 8-4=2k\]
Now, on simplifying it further we get,
\[\Rightarrow 2k=4\]
Let us now divide with 2 on both the sides
\[\Rightarrow k=\dfrac{4}{2}\]
Now, on further simplification we get,
\[\therefore k=2\]

Note:
Instead of considering that the common difference of the consecutive terms remains constant we can also solve this by considering that if three terms a, b, c are in arithmetic progression then we have the condition that \[2b=a+c\]which on further simplification gives the result.
It is important to note that while calculating the difference of the consecutive terms we need to make sure that while equating in both the differences we need to consider second term minus first term. Because in one case if we consider second term minus first term and in the other case if we consider the first term minus second then it will be completely incorrect.