Question

Which term of the sequence \[25,24\dfrac{1}{4},23\dfrac{1}{2},22\dfrac{1}{2},......\] is the first negative term?

Answer 1

Hint: Here we check if the sequence given to us is an AP by checking if the common difference between the terms is the same or not. Then taking the first term and common difference in AP we calculate the value of the $n^{th}$ term using the formula of the $n^{th}$ term of an AP and we take that term to be zero because the first negative term will come just after zero as the sequence is in decreasing order.
* In an arithmetic progression, the first term is a, a common difference is d and n is the number of terms, then, $n^{th}$ term is given by \[{a_n} = a + (n - 1)d\].
* Mixed fraction can be converted to normal fraction by \[a\dfrac{b}{c} = \dfrac{{ac + b}}{c}\]

Complete step-by-step answer:
We are given the sequence \[25,24\dfrac{1}{4},23\dfrac{1}{2},22\dfrac{1}{2},......\]
Here we find the difference between two consecutive terms and check if the difference is the same then the sequence is an AP.
For first and second term:
Convert the term from mixed fraction to normal fraction in RHS.
\[ \Rightarrow 24\dfrac{1}{4} - 25 = \dfrac{{24 \times 4 + 1}}{4} - 25\]
\[ \Rightarrow 24\dfrac{1}{4} - 25 = \dfrac{{97}}{4} - 25\]
Write the fraction in RHS as a decimal number.
\[ \Rightarrow 24\dfrac{1}{4} - 25 = 24.25 - 25\]
\[ \Rightarrow 24\dfrac{1}{4} - 25 = - 0.75\]
For second and third term:
Similarly, convert the term from mixed fraction to normal fraction in RHS.
 \[ \Rightarrow 23\dfrac{1}{2} - 24\dfrac{1}{4} = \dfrac{{23 \times 2 + 1}}{2} - \dfrac{{24 \times 4 + 1}}{4}\]
\[ \Rightarrow 23\dfrac{1}{2} - 24\dfrac{1}{4} = \dfrac{{47}}{2} - \dfrac{{97}}{4}\]
Write the fraction in RHS as a decimal number.
\[ \Rightarrow 23\dfrac{1}{2} - 24\dfrac{1}{4} = 23.5 - 24.25\]
\[ \Rightarrow 23\dfrac{1}{2} - 24\dfrac{1}{4} = - 0.75\]
Since the difference is the same between two consecutive terms, the sequence is an AP.
Now first term of the sequence \[a = 25\]
Common difference \[d = - 0.75\]
Now we assume that $n^{th}$ term is equal to zero.
Using the formula for $n^{th}$ term of an AP we have
\[ \Rightarrow 0 = a + (n - 1)d\]
Substitute the value of a and d in the equation.
\[ \Rightarrow 0 = 25 + (n - 1)( - 0.75)\]
Multiply the terms on RHs of the equation.
\[ \Rightarrow 0 = 25 - 0.75n + 0.75\]
\[ \Rightarrow 0 = 25.75 - 0.75n\]
Shift the constant value to one side of the equation.
\[ \Rightarrow 0.75n = 25.75\]
Divide both sides by 0.75
\[ \Rightarrow \dfrac{{0.75n}}{{0.75}} = \dfrac{{25.75}}{{0.75}}\]
Cancel the same terms from numerator and denominator.
\[ \Rightarrow n = 34.33\]
Since n is the number of the term of AP, it cannot be in decimal form.
Therefore for the value of \[n = 34\]
\[{a_{34}} = a + (34 - 1)d\]
Substituting the values of a and d
\[{a_{34}} = 25 + 33 \times ( - 0.75)\]
\[{a_{34}} = 25 - 24.75\]
\[{a_{34}} = 0.25\] This is positive.
So, we calculate the next term by adding the common difference -0.75
\[{a_{35}} = 0.25 + ( - 0.75)\]
\[{a_{35}} = 0.25 - 0.75\]
\[{a_{35}} = - 0.5\]
So, the first negative term is $35^{th}$ term of the sequence which is equal to -0.5.

Note: Students many times try to find the value by long calculation where they add the common difference to each term to get the next term which is a really long process, so we should avoid it. Also, many students make mistakes while converting the mixed fraction as they multiply the whole number by numerator and add the denominator which is wrong, always multiply the whole number to denominator first and add the denominator to it which forms the new numerator.