Help with a pattern/question

[jw_gino]

Is there a simplified way to answer this question? if so, how?


What is the sum of the integers in the table?

1 2 3 4 5 6 7
-2 -4 -6 -8 -10 -12 -14
3 6 9 12 15 18 21
-4 -8 -12 -16 -20 -24 -28
5 10 15 20 25 30 35
-6 -12 -18 -24 -30 -36 -42
7 14 21 28 35 42 49

Thank you in advance?

[jnelson0612]

Is there a simplified way to answer this question? if so, how?


What is the sum of the integers in the table?

1 2 3 4 5 6 7
-2 -4 -6 -8 -10 -12 -14
3 6 9 12 15 18 21
-4 -8 -12 -16 -20 -24 -28
5 10 15 20 25 30 35
-6 -12 -18 -24 -30 -36 -42
7 14 21 28 35 42 49

Thank you in advance?

Interesting question!

Okay, let's first just look at the positive numbers:
1 2 3 4 5 6 7
3 6 9 12 15 18 21
5 10 15 20 25 30 35
7 14 21 28 35 42 49

Look at how each of these rows shows a multiple of some number. We have:
a) multiples of 1: 1(1+2+3+4+5+6+7)
b) multiples of 3: 3(1+2+3+4+5+6+7)
c) multiples of 5: 5(1+2+3+4+5+6+7)
d) multiples of 7: 7(1+2+3+4+5+6+7)

So what is (1+2+3+4+5+6+7)? 28. We have (1 + 3 + 5 + 7) times 28, or 16 * 28 for the positive numbers.

Okay, let's crunch through the negatives:
-2 -4 -6 -8 -10 -12 -14
-4 -8 -12 -16 -20 -24 -28
-6 -12 -18 -24 -30 -36 -42

Again, using multiples:
a) -2(1+2+3+4+5+6+7, which I now know is 28)
b) -4 (28)
c) -6(28)

So, (-2 - 4 - 6) 28, or -12 * 28.

Thus, I have 16*28 + (-12)28, or (16-12)28. 4*28 is 112.

[lyl9021]

1 2 3 4 5 6 7
-2 -4 -6 -8 -10 -12 -14
3 6 9 12 15 18 21
-4 -8 -12 -16 -20 -24 -28
5 10 15 20 25 30 35
-6 -12 -18 -24 -30 -36 -42
7 14 21 28 35 42 49

s0=1+2+3+4+5+6+7=(1+7)*7/2=28
s1=-2*(1+2+3+4+5+6+7)=-2*s0
s2=3s0
s4=-4s0
s5=5s0
s6=-6s0
s7=7s0
s=(1-2+3-4+5-6+7)s0=[(1+3+5+7)-(2+4+6)]s0=4s0=4*28=112

[jw_gino]

Thank you

[tim]

nice approach!