Assume that two athletes sign 10-year contracts that pay out a total of $100 million over the life of the contracts. One contract will pay the $100 million in equal installments over the 10 years. The other contract will pay the $100 million in installments, but the installments increase 5% per year. Which athlete received the better deal?
+2440 Thank you for the reply. I actually did find these numbers, but not directly. I did this by creating a spread sheet starting with a value of X and incremented each successive value of X by multiplying by 1.05. I also kept a running total. When the 10th value = 100,000,000, I knew that was the correct starting value. (The 10th value was actually = 100,000,000.04 – a rounding error, I guess). It took about a half-hour to zero in on the starting value.
What I didn’t know how to do was find the starting value directly. Even though it is not an interest rate problem, I tried using the present value formula, with the interest rate set at 105% per year with a FV set at 100,000,000. . . . Nope! Not even close. I tried adjusting the “interest rate” by using a harmonic average, geometric average, and the median. Nope!
Now that you‘ve pointed out this is a geometric series, I read the wiki on this subject and found the formula relating to this.
https://en.wikipedia.org/wiki/Geometric_series#Economics
\(s=a{\frac {1-r^{n}}{1-r}}\)
After solving the formula for (a), the solution instantly popped out, like a “Where’s Wally” character. 
Thanks guest. I always enjoy finding Wally.
Isn't obvious to you which one has a better contract?
1) The first athlete simply gets 1/10 of $100 million, or $10 million per year for ten years.
2) The second athlete will begin with $10 million the 1st year, $10.5 million the 2nd year, $11.025 million the 3rd year and so on for a total of $125,778,925.36 over the ten-year period.
But the best comparison between their two contracts is to find the PV, or present value, of their contracts as of today. Using that method:
1) The PV of the first athlete discounted at 5% compounded annually=$77,217,349.29 as of today's money.
2) The PV of the second athlete is of course =$100,000,000 as of today's money.
3) The difference between them being=$100,000,000 - $77,217,349.29 =$22,782,650.71
+2440 Jesus Christ! where is Nauseated?
The dumbest of athletes would probably know the first one is a better deal. If for no other reason they could afford a financial advisor to inform them.
Lately, these Batshit-Stupid answers have become so numerous, it's causing a major out-break of Contagious Dumbness Disease.
Nauseated, where in Hades's shadow are you?!
Nauseated is an amazingly popular troll, especially with the chicks.
Can you explain why the first choice is a better deal?
+2440 OMG!! I posted this as a guest. I’m glad I wasn’t too foul in my choice of words. Stupid and dumb really do annoy me --especially when I am the propagator of it.
I still stand by my answer, though. The reason is these contracts do not say anything about paying interest, only an increase in installments. Inflation makes today’s money worth more than tomorrow’s money. The first contract pays faster so it is the better deal.
+118608 Hi Ginger
You know I saw you logged on. Then I saw you had logged off.
Next these 'anonymous' posts appeared. I knew immediately that you had posted them. I wondered why you did that but I just guessed that you were of 2 minds about claiming your written creation. :)
I would have known that they were your words even if you had not logged on. It is not always as easy to hide as people think. :)
Anyway it is nice you are here, I am nostalgic too:)
+2440 That is amazing you could tell it was me. I’ve made only a few posts using my account name and some as a guest. Some were snarky, trolling-posts and were deleted – they were not Scooby-snack-worthy.
My popping on and off happened because I logged on through a university library network. I was going to post a question (details below) because my “private tutor” was off the grid. When I realized I couldn’t log on, my Irish temper took over and I flamed the post with two of Naus’ thermonuclear grenades of humor and fire. It did cheer me up. It was still anonymous 10 hours later when I checked for responses. It’s not beyond me to post something like this with my name attached, but it’s kind of dumb to do it and expect anything except an in-kind retort.
When I was 7-years-old, I learnt it was much more fun to sneak a cookie than to ask. My mom and dad were amused by behavior, but my dad, being somewhat of a practical joker, stuffed a jack-in the-box, springy, caterpillar in the cookie jar. On top of that, he hid in the pantry with a cookie monster mask. I managed not to drop the cookie jar when the caterpillar popped out, but when cookie monster jumped out of the pantry, the last thing I remember before fainting, was throwing the cookie jar at him, which he caught (it was Rubbermaid—my mother is a smart cookie). I’m sure my father was more scared than I was at that point.
I didn’t eat any cookies for weeks afterward. It was at least a year before I again started sneaking cookies, but I carefully and slowly opened the jar. To this day, after more than 20 years, I still half believe something will jump out whenever I open a cookie jar. 
+118608 Your posts are unmistakable Ginger. :)
If you have had some posts disappear someone must have thought they were worth while scooby snacks, but to be fair scooby snacks come in many different flavours. :)
Your Dad sounds like a LOT of fun. What a lucky child you were. 
+2440 It’s no surprise Naus is popular with many “chicks” on this forum, and probably more than a few despise him too. Now that the cat is out of the bag, so to speak, I confess that there is one chick who dearly loves him. Incase it’s not obvious, that chick is me. I have a bachelor's degree now, and I’m working toward my masters in art. That would not have happened except for Naus. He tutored me through two terms of required mathematics and science courses. I even took an elective statistics class. I didn’t just pass these classes, I aced them!
Naus is accomplished in many of the arts too. His knowledge in literature and music are particularly astonishing. Once I told him I had a research term paper for an art history class due in less than 10 days. Two days later he sent me a 12-page outline comparing and contrasting Dadaism and kitsch art in general and specific forms. His details and annotations were so well defined I finished the term paper in five days. I could have finished in four, except uncontrollable laughter came upon me because three of the pages were written in his LancelotLink persona that included detailed contrasts of the banana daiquiri-drinking chimps to the beer-drinking gorillas, along with human Buddha-bellies as a control group. My professor, who is very allergic to giving an “A” to anyone, gave one to me along with an accolade of excellence. (I still made only a “B” in the class though)
As you may have guessed by my avatar, my favorite of his personas is Lancelotlink.
But whether LancelotLink, Nauseated, the Toll-collecting Troll, or his many other personas –many of which are unknown on here, he is my dear friend and I love him and miss him dearly.
+118608 Hello GingerAle it is nice to see you 'having a go' :)
You are certainly correct that the one who gets the money quicker gets the better deal.
This question is ambiguous but the way I interprete is that the first one gets 100000 each year for 10 years.
The second one gives 100000 the first year then more each subsequent year until the whole 1 million dollars has been received.
So the second one gives the money faster and is the better deal. :)
+2440
It was this phrase, “. . . pay out a total of $100 million over the life of the contracts,” that made me think the two contracts had the same final dollar value. So the only difference in real value would be the inflation rate.
“Ambiguous.” One of Naus’ favorite words. It’s true though, now I do not know which way is correct.
+118608 Oh I have only just seen you last post.
Congratulation on your great achievements. !!!
I am also very pleased that you found a great tutor here. Perhaps Nauseated will return. :)
I am sure he is pleased by your success and that he appreciates your gratitude :)
+2440 Thank you Miss Melody. You are very kind.
Naus has always spoken very highly of you. And of CPhill, Alan, Heureka, Rom, Bertie and Geno, too, although he never refers to him by his name. Always the troll he is. 
I was making a banana daiquiri and the lid popped off the blinder. What a mess!! 
My cat is helping me mop it up.
+118608 LOL
I bet the cat did a good job.
Is it stumbling around with head in paws or is it sleeping it off in a cosy corner?
+2440 Oh, I hadn’t put the rum in it, yet. This banana daiquiri recipe calls for 80 ml of rum which is too much for me –and my cat :). I drizzle the rum (a few dozen imperial minims at a time) on top as I drink it. I drink this slowly otherwise it gives me a “brain freeze” headache. My cat seems immune to these, but he wouldn’t be immune to the alcohol in the rum, though he probably could handle it better than me.
I am of very Irish heritage but I didn’t inherit much of a tolerance for alcohol. I would get a light buzz from just a few sips of communion wine. In the scheme of life, this is probably a good thing.
#2 GingerAle:
You were thinking of an "escalator clause" of 5% in the second contract. If that is what you thought the questioner meant, then it is very easy to calculate:
Based on a total of $100 million the 1st. payment would be=$7,950,457.50. This amount would increase or escalate by 5% per year. The 10th and last payment would be=$12,333,769.04. The total of the ten escalated payments would be exactly $100 million. You sum them up as a geometric series, with the first term being $7,950,457.50, the ratio being 1.05, and the number of terms being 10.
+2440 Thank you for the reply. I actually did find these numbers, but not directly. I did this by creating a spread sheet starting with a value of X and incremented each successive value of X by multiplying by 1.05. I also kept a running total. When the 10th value = 100,000,000, I knew that was the correct starting value. (The 10th value was actually = 100,000,000.04 – a rounding error, I guess). It took about a half-hour to zero in on the starting value.
What I didn’t know how to do was find the starting value directly. Even though it is not an interest rate problem, I tried using the present value formula, with the interest rate set at 105% per year with a FV set at 100,000,000. . . . Nope! Not even close. I tried adjusting the “interest rate” by using a harmonic average, geometric average, and the median. Nope!
Now that you‘ve pointed out this is a geometric series, I read the wiki on this subject and found the formula relating to this.
https://en.wikipedia.org/wiki/Geometric_series#Economics
\(s=a{\frac {1-r^{n}}{1-r}}\)
After solving the formula for (a), the solution instantly popped out, like a “Where’s Wally” character.
Thanks guest. I always enjoy finding Wally.
GingerAle: Since you are of Irish Heritage, here is some typical Irish Humour that you could share with family and friends:
A MOTHER'S LETTER
Dear Son.
Just a few lines to let you know that I’m still alive. I’m writing this slowly because I know that you can’t read fast. You won’t know the house when you come home, we have moved.
About your father. He’s got a lovely new job. He has 500 men under him. He cuts grass at the cemetery. Your sister Mary had a baby this morning. I haven’t found out yet whether it’s a boy or a girl so I don’t know if you are an aunt or an uncle.
I went to the doctors on Thursday and your father came with me. The doctor put a small tube in my mouth and told me not to talk for 10 minutes. Your father offered to buy it from him. Your uncle Patrick drowned last week in a vat of Irish whiskey at the Dublin brewery. Some of his workmates tried to save him but he fought them off bravely. They cremated him and it took three days to put the fire out.
It only rained twice this week, first for 3 days then for 4 days. We had a letter from the undertaker. He said if the last payment on your grandmother’s plot wasn’t paid in 7 days, up she comes
Your loving mother.
P. S. I was going to send you 5 pounds but I have already sealed the envelope.
