Question

Please help!

Solve the equation for x.

Answer 1

`\bf \displaystyle\sum\limits_(n=1)^(6)~(x-3n)=-3\implies \sum\limits_(n=1)^(6)~x-\sum\limits_(n=1)^(6)~3n=-3\implies \sum\limits_(n=1)^(6)~x-3\sum\limits_(n=1)^(6)~n=-3 \\\\\\ 6(x)~~-~~3\left[ \cfrac{6(6+1)}{2} \right]=-3\implies 6x~~-~~3[3(7)]=-3\implies 6x - 63 = -3 \\\\\\ 6x=60\implies x=\cfrac{60}{6}\implies x=10`

Answer 2

c. 10

Explanation:

First of all, the sum can be simplified.

`\sum\limits^6_(n=1) {(x-3n)}=\sum\limits^6_(n=1) {x}-3\sum\limits^6_(n=1) {n}\\\\=6x-3\cdot(6\cdot 7)/(2) =6x-63`

This result can be used in the equation ...

6x -63 = -3

6x = 60 . . . . . add 63

x = 10 . . . . . . . divide by 6

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Comment on the ∑n

The sum of numbers 1 to n is n(n+1)/2. We have used that fact here. For this problem, the sum is short enough you can simply add it up: 1+2+3+4+5+6 = 21.