Question

What is the sum of this arithmetic series? 586+564+542+...+212

Answer 1

Basically it's asking for the sum of 212 + 216 + 220 + ..... 586

Each number is 22 more than the previous one.

Therefore the sum will be 212 + (212+22) + (212+22*2) + (212+22*3)

the amount of numbers from 212 through 586 is 18.

Therefore we will need 212 plus 212 * 17 = 3,816 *******************

We will also need all those 22's.

We must add 22 *1 plus 22*2 plus 22*3 ..... plus 22*17

Which equals 22 + 44 + 66 + 88 ... 374

Which totals 3,366 *****************

So, we total 3,816 + 3,366 which equals 7,182

Explanation:

Answer 2

`\displaystyle\bf\\Sum=586+564+542+...+212\\\\Sum=212+234+256+...+586\\\\\textbf{We calculate the number of terms (n):}\\\\n=(586-212)/(22)+1=(374)/(22)+1=17+1=18\\\\\boxed{\bf~n=18~terms}\\\\Sum=(n(586+212))/(2)\\\\Sum=(18* 798)/(2)\\\\Sum=9*798\\\\\boxed{\bf~Sum=7182}`