Question

solve the following

$7 \frac{1}{7} \div3 \frac{4}{7} + { (1 - \frac{1}{2} - 1 \frac{1}{4} ) \times (1 \frac{1}{3} + 1 \frac{3}{9} )}$

Answer 1

Answer:

Step-by-step explanation:

Remaining solve you

Answer 2

Answer:

Step-by-step explanation:

The first thing you should do is write out the first few terms, and sum them up and see if you see any patterns emerging. Is there anything you can generalize? Can you prove that your pattern will hold?

13+16+110+115⋯13+16+110+115⋯

Lets work out the partial sums. That is, work left to right and write down what you have so far and what you get when you add one more term.

13,12,35,23⋯13,12,35,23⋯

Interesting, every fraction reduces to something pretty simple.

What if we didn’t put it in lowest terms. What if we did this?

13,24,35,46⋯13,24,35,46⋯

Curious! What is going on?

Lets get deeper into the math.

1+2+3⋯n=12n(n+1)1+2+3⋯n=12n(n+1)

We can rewrite your problem

∑n=220172n(n+1)∑n=220172n(n+1)

But we can make it simpler!

2n(n+1)=2n−2n+12n(n+1)=2n−2n+1

Which means

∑n=220172n(n+1)=∑n=22017(2n−2n+1)∑n=220172n(n+1)=∑n=22017(2n−2n+1)

Now write out the first few terms of that… and what do you see?

1−23+23−24+24⋯−22017+22017−220181−23+23−24+24⋯−22017+22017−22018

A whole lot of terms cancel leaving just the first and the last term.

1−22018