Question

What is the value of $1 - 2 + 3 - 4 + 5 - ... + 101$?
A) 51
B) 55
C) 110
D) 111

Answer 1

Here, we will first separate the terms with negative signs and positive signs. Then we use the formula of first $n$ odd numbers, ${n^2}$ in the first part of the obtained expression and the formula of $n$ natural numbers, in the second part of the obtained expression. Then we will simplify it to find the required value.

Complete step by step solution: We are given that $1 - 2 + 3 - 4 + 5 - ... + 101$.

First, we will write the terms with negative signs and positive signs separately in the above expression.

$\Rightarrow \left( {1 + 3 + 5 + ... + 101} \right) - \left( {2 + 4 + 6 + ... + 100} \right)$

On taking 2 common from the second part of the above expression, we get

$\Rightarrow \left( {1 + 3 + 5 + ... + 101} \right) - 2\left( {1 + 2 + 3 + ... + 50} \right){\text{ ......eq.(1)}}$

We have seen that the first part of the above equation is a sum of first $n$ odd numbers and the second part is a sum of first $n$ natural numbers.

Since there are 51 terms in the first sum of the above expression, then the value of $n$ is 51.

We will now use the formula to the sum of odd numbers is ${n^2}$ in the first part of the above equation by replacing 51 for $n$, we get

$$\Rightarrow {51^2} \\\ \Rightarrow 51 \times 51 \\\ \Rightarrow 2601 \\\$$

Substituting the above value in the equation (1) to simplify it, we get

$\Rightarrow 2601 - 2\left( {1 + 2 + 3 + ... + 50} \right)$
Using the formula of sum of $n$ natural numbers, $1 + 2 + 3 + ... + n = \dfrac{{n\left( {n + 1} \right)}}{2}$ in second part of the above expression where $n = 50$, we get

$$\Rightarrow 2601 - 2\dfrac{{50\left( {50 + 1} \right)}}{2} \\\ \Rightarrow 2601 - 50\left( {50 + 1} \right) \\\ \Rightarrow 2601 - 50\left( {51} \right) \\\ \Rightarrow 2601 - 2550 \\\ \Rightarrow 51 \\\$$

Thus, the value of $1 - 2 + 3 - 4 + 5 - ... + 101$ is 51.

Hence, option A is correct.

Note: While solving these types of problems, students should use the first, second and third term to find the general format of this series. The key concept of this question is to find the formulas of finite series. The numbers in the sum $1 + 3 + 5 + ... + 101$ is 50 instead of 51 can be confusing sometimes.