Question

A series whose $n^{th}$ term is \[\left( {n/x} \right) + y\], the sum of r terms will be
1) \[(r(r + 1)/2x) + ry\]
2) \[r(r - 1)/2x\]
3) \[(r(r - 1)/2x) - ry\]
4) \[(r(r + 1)/2y) + rx\]

Answer 1

Hint: the given simple problem can be solved easily as $n^{th}$ term is given in the problem, we need to find the sum of rth term for that we need to find at least first 3 terms and rth term and we can obtain by giving the value of n as 1, 2, 3 and so on r. Then by adding all the terms she gets the required solution.

Complete step by step answer:
Now let us consider the given data
Since $n^{th}$ term of the series is given as
\[{a_n} = \dfrac{1}{x} + y\]
Now let us find the first term by giving the value of n as 1 we get
\[{a_1} = \dfrac{1}{x} + y - - - \left( 1 \right)\]
To get the second term of the series given nis equal to 2 we get
\[{a_2} = \dfrac{2}{x} + y - - - \left( 2 \right)\]
Similarly put n as 3 we get the third term of the series and is given by
\[{a_3} = \dfrac{3}{x} + y - - - \left( 3 \right)\]
Now to get rth term replace n by r in the given series we get
\[{a_r} = \dfrac{r}{x} + y - - - \left( 4 \right)\]
Since we need to calculate the sum of r terms of the series, we can obtain it by adding equations 1, 2, 3, and 4 and is given by
Sum of r terms of the given series is \[ = \dfrac{1}{x} + y + \dfrac{2}{x} + y + \dfrac{3}{x} + y + - - - \dfrac{r}{x} + y\]
Since we have the term y r number of times so the above expression can be written as
Sum of r terms of the given series is \[ = \dfrac{1}{x} + \dfrac{2}{x} + \dfrac{3}{x} - - - \dfrac{r}{x} + ry\]
Taking \[\left( {\dfrac{1}{x}} \right)\] as a common factor we get
Sum of r terms of the given series is \[ = \left( {\dfrac{1}{x}} \right)\left( {1 + 2 + 3 + - - - + r} \right) + ry\]
We know that the sum of r term of a natural numbers \[1 + 2 + 3 + - - - + r\] by using this expression in the above step we get
Sum of r terms of the given series is \[ = \left( {\dfrac{1}{x}} \right)\left( {\dfrac{{r(r + 1)}}{2}} \right) + ry\]
Sum of r terms of the given series is \[ = \left( {\dfrac{{r(r + 1)}}{{2x}}} \right) + ry\]

So, the correct answer is “Option 1”.

Note: A series can be highly generalized as the sum of all the terms in a sequence. However, there has to be a definite relationship between all the terms of the sequence.
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis.