Question

There are $ m $ arithmetic means between $ 5 $ and $ - 16 $ such that the ratio of the $ 7th $ mean to the $ (m - 7)th $ mean is $ 1:4 $ . Find $ m $ .

Answer 1

Hint: As we know that an arithmetic progression or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant. Here in this question we have to find the value of $ m $ , an unknown number. We know the formula of common difference if a number is inserted between two numbers is $ d = \dfrac{{b - a}}{{n + 1}} $ where $ n $ is the unknown term, $ a = $ first term and $ d $ is the common difference. We will first find the common difference between the two terms of the given series and then we will substitute the values in the formula.

Complete step by step solution:
As the per given question we have the sequence: $ 5,m $ and $ - 16 $ .
We have $ a = 5,b = - 16 $ and $ n = 20 $ . So by substituting the values in the common difference of arithmetic sequence formula:
 $ d = \dfrac{{ - 16 - 5}}{{m + 1}} = \dfrac{{ - 21}}{{m + 1}} $ .
So the new ratio is $ \dfrac{{1 + 7d}}{{1 + (m - 7)d}} = \dfrac{1}{4} $ .
Now by cross multiplication we will solve this:
 $ 4 + 28d = 1 + md - 7d \\
\Rightarrow 4 - 1 + 28d + 7d = md
 $
On further solving we have, $ 3 + 35d = md $ . Now we will put the value of $ d $ in the equation,
 $ \Rightarrow 3 + 35 \times \dfrac{{ - 21}}{{m + 1}} = m\left( {\dfrac{{ - 21}}{{m + 1}}} \right) $ . We will solve it now:
$ \dfrac{{3m + 3 + 35 \times - 21}}{{m + 1}} = \dfrac{{ - 21 \times m}}{{m + 1}} $ .
 $ 3m + 3 - 735 = - 21m \\
\Rightarrow 3m + 21m = 732 $ . It gives us the value of
$ m = \dfrac{{732}}{{24}} = 30.5 $ .
Hence the value of $ m $ is $ 31(approx) $ .
So, the correct answer is “ $ 31(approx) $ ”.

Note: We should be aware of the arithmetic sequence and their formula before solving this kind of question. We should carefully substitute the values and solve them. Also we should know that to find the sum of the given arithmetic sequence by adding the first and last term and then divide the sum by two, this is also a formula when the first and the last term is given in the question. And then the sum of the sequence will be the number of terms multiplied by the average number of terms in the sequence. The formula can be written as $ S = \dfrac{n}{2}(a + l) $ where the number of terms and $ a,l $ are first and last terms.