1. When Yin attempts to add all the positive integers from 1 through 99, he accidentally skips a number and gets the sum 4909. Which number did Yin skip?
2. Compute the sum of \(\[\left(1 - \frac 16\right) + \left(\frac12 - \frac17\right) + \left(\frac13 - \frac18\right) + \cdots + \left(\frac{1}{95} - \frac{1}{100}\right).\]\) Express your answer as a decimal to the nearest tenth.
3. Compute the product \(\[(-39 + 13)\cdot(-36 + 12) \cdot(-33 + 11) \cdot(-30 + 10) \cdots (33 -11)\cdot (36 - 12) \cdot (39 - 13).\]\)Where the first summand in each factor is increasing by 3 and the second summand is decreasing by 1.
1. When Yin attempts to add all the positive integers from 1 through 99, he accidentally skips a number and gets the sum 4909. Which number did Yin skip?
The formula for the sum of numbers 1 + 2 + 3 • • • + n is (n)(n+1) / 2
(99)(100) / 2 = 4950 If Yin got 4909 he omitted (4950 – 4909) = 41
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+23246 #2: The positive terms are:
1 + 1/2 + 1/3 + 1/4 + 1/5 + 1/6 + 1/7 + 1/8 + ... + 1/93 + 1/94 + 1/95
The negative terms are:
- 1/6 - 1/7 - 1/8 - ... -1/93 - 1/94 - 1/95 - 1/96 - 1/97 - 1/98 - 1/99 - 1/100
Cancelling the equivalent positive and negative terms, we are left with:
1 + 1/2 + 1/3 + 1/4 + 1/5 - 1/96 - 1/97 - 1/98 - 1/99 - 1/100
Now, it's calculator time!
#3: If you write out the factors, you have, for one of the factors, (-12 + 12) which is 0.
Therefore, the product is 0.
