QUESTION 4 d. Fidelity Investments publishes a magazine a few times a year. In 2

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QUESTION 4 d. Fidelity Investments publishes a magazine a few times a year. In 2008 the President of Fidelity Investments was Steven P. Akin. Here is what he said on his President's Note page of the magazine Doing a quick calculation, I found that a sum of $10,000 in 1993 had the same buying power as S14,867 does in 2008. In other words, $10,000 has lost almost half its real value in the past 15 years. What do you think about the mathematics of what Mr. Akin said? e. You record a two-digit number incorrectly by transposing its digits. Show that the difference between the correct number and the transposed number yields a whole number when divided by nine. f. Same question as (e) but assume that you transpose two adjacent digits in a three-digit number. g. Notice that 1-1 1+3-4 1+3+5-9 1+3+5+7 16 Show that the sum of the first n odd integers equals n h. Find a compact expression for the sum of the first n integers

Explanation / Answer

d) If $10,000 in 1993 had same buying value as $14,867 in 2008, then they will be able to buy same quantity, say x, of a certain product. Now if in 2008, $14,867 buy x units then $10000 will buy (10,000/14,687)*x = 0.67x units of the same product.  

Thus buying capacity of $10000 has reduced from x to 0.67x ie. buying value has reduced to 33%.Thus $ 10,000 has lost only 33% of its real value in past 15 years.

e) Let x and y be any number between 0 and 9 such that 10x+y is the correct number. Then the incorrect number will be 10y+x.

Hence the difference between two numbers will be (10x+y)-(10y+x) = 9x-9y = 9( x-y ), which is alway devisible by 9.

g) The 1st n odd integers are given by,

(1+(1+2)+(1+4)+(1+6)+...............+(1+2(n-1)) = n+2(1+2+.....(n-1)) = n + 2(n-1)n/2 (Since, sum of 1st n-1 integers) = n+ n^2 - n = n^2.

h) Let T(n) be the sum of first n integers,

then T(n) + T(n) =1 + 2 +..............+ (n-1) + n + n + (n-1) + .......+ 2 + 1

which can be rewritten as 2 T(n) = (1+n) + ( 2+ (n-1)) +...........+ ((n-1) +2) + ( n+1) = (n+1)+(n+1) +........(n+1) +(n+1) = n(n+1)

Thus T(n) = n(n+1)/2

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