What is the value of $ 2013-2009+2005-2001 + \cdots + 29-25 $?

Question

Calculate the value of: $$ 2013-2009+2005-2001 + \cdots + 29-25 $$ Ok I tried to answer this first by arranging the numbers: $$ (2013-2009)+(2005-2001)+ \cdots + (29-25) $$ so that the answer in each bracket is 4. How do I calculate the number of times 4 appears? That is what I'm stuck on.

Answer 1

Another approach :

Rewrite: $$ 2013-2009+2005-2001+\cdots+29-25 $$ as $$ (2013+2005+\cdots+29)-(2009+2001+\cdots+25). $$ the first and second series are arithmetic progressions (AP), where the difference between the consecutive terms for each AP is $-8$. Now, take a look the first AP. You have $a_1=2013$ and $a_n=29$, therefore $$ a_n=a_1+(n-1)b\quad\Rightarrow\quad29=2013+(n-1)(-8)\quad\Rightarrow\quad n=249. $$ Similarly for the second AP. You have $c_1=2009$ and $c_n=25$, therefore $$ c_n=c_1+(n-1)d\quad\Rightarrow\quad25=2009+(n-1)(-8)\quad\Rightarrow\quad n=249. $$ Now, you can conclude that you will have $249$ terms of $4$. Thus, $$ (2013-2009)+(2005-2001)+\cdots+(29-25)=4+4+\cdots+4=249\cdot4=\boxed{\Large\color{blue}{996}} $$

Answer 2

Let there are $n$ terms

So the $n$th term $=2013=\displaystyle2013=25+(n-1)4\implies n=498$

So, there are $\left\lfloor\dfrac n2\right\rfloor$ such pairs

Had $n$ been odd, there $\left\lfloor\dfrac n2\right\rfloor$ such pairs $+1$ unpaired number

Answer 3

Observe the first term in each bracket: $2013, 2005, 1997, ..., 29$. The common difference is $8$. So $2013$ must be $29$ plus $8$ times of the number of terms between them (including $2013$, excluding $29$). Thus, there are $\frac{2013 - 29}{8}$ terms between them. We have to count $29$ itself, so we add $1$ to this result.

There are therefore $\frac{2013 - 29}{8} + 1$ pairs of numbers that form the number $4$. Their sum would simply be:

$$4\cdot\left(\frac{2013 - 29}{8} + 1\right) = 996$$

Answer 4

You were on right track by your solution the best method is find out the number of terms by ignoring the signs you will have

2013 = 25+(n-1)*4

n= 498

so the number of fours will be 498/2 = 249

so sum = 249 * 4 = 996