Question

Please explain to me how to do this

Answer 1

#43

• 1+2+3+..+200
• We may observe. 200+1=201,199+2=201,198+3=201

So

200/2=100 pairs of 201

Sum

• 100(201)=20100

#44

• 1+2+3+..+400

400/2=200 pairs of 401

Sum

• 200+401
• 80200

#45

• 1+2+3+..+800

800/2=400 pairs of 801

Sum

• 400(801)
• 320400

#46

• 1+2+3+..+2000

2000/2=1000 pairs of 2001

Sum

• 2001(1000)
• 2001000

Answer 2

43. 20,100

44. 80,200

45. 320,400

46. 2,001,000

Explanation:

Gauss's method

General formula for the sum of the first n integers:

`1+2+3+ \dots +n=(1)/(2)n(n+1)`

Question 43

Given sum:

• 1 + 2 + 3 + ... + 200

Therefore, n = 200.

Substitute the value of n into the formula:

`\implies (1)/(2)(200)(200+1)=100 \cdot 201=20100`

Question 44

Given sum:

• 1 + 2 + 3 + ... + 400

Therefore, n = 400.

Substitute the value of n into the formula:

`\implies (1)/(2)(400)(400+1)=200 \cdot 401=80200`

Question 45

Given sum:

• 1 + 2 + 3 + ... + 800

Therefore, n = 800.

Substitute the value of n into the formula:

`\implies (1)/(2)(800)(800+1)=400 \cdot 801 =320400`

Question 46

Given sum:

• 1 + 2 + 3 + ... + 2000

Therefore, n = 2000.

Substitute the value of n into the formula:

`\implies (1)/(2)(2000)(2000+1)=1000 \cdot 2001=2001000`