On the first Sunday of 2003, Rizzo and Frenchie start a chain letter, each of them sending five letters (to ten different friends between them). Each person receiving the letter is to send copies to five new people on the Sunday following the letter’s arrival. After the first seven Sundays have passed, what is the total number of chain letters that have been mailed? How many were mailed on the last three Sundays?
The way I solved it was to add them together to find the number of letters that have been mailed.
a)$2 \cdot 5 + 2 \cdot 5^2 + 2 \cdot 5^3 + 2 \cdot 5^4 + 2 \cdot 5^5 + 2 \cdot 5^6 + 2 \cdot 5^7 = 195310$
b) $2 \cdot 5^5 + 2 \cdot 5^6 + 2 \cdot 5^7 = 193750$
Did I do the right thing?
Best Answer
Yes you are correct. Another way to see it is by counting the number of chain letters that have been mailed from the ones Rizzo's or Frenchie's sent individually. From Rizzo's side we have
$$\displaystyle\sum\limits_{i=1}^{7}5^i=5^1+5^2+\cdots+5^7$$
total numbers of chain letters sent, where $i$ denotes the $i$-th sunday. Similarly for Frenchie's we have the same amount $$\displaystyle\sum\limits_{i=1}^{7}5^i=5^1+5^2+\cdots+5^7.$$
So for (a) we have $2\displaystyle\sum\limits_{i=1}^{7}5^i=2(5^1+5^2+\cdots+5^7)=195,310.$ For (b) we want the number of chain letters starting from the last three sundays, so we want the 5-th, 6-th, and 7-th sundays. Set $i=5$, so $2\displaystyle\sum\limits_{i=5}^{7}5^i=2(5^5+5^6+5^7)=193,750.$