Kevin’s dog Amadeus..., & The 11th term of an A.P is -17...

[Altair]

Kevin’s dog Amadeus..., & The 11th term of an A.P is -17...

Hey guys thanks for taking the time for looking at this. Instead of posting a bunch of small threads after trying to work all these out i posted the ones in need help with, so here it is.

1) Kevin’s dog Amadeus like two kinds of canned dog food. Gourmet Dog and Chow Hound. Gourmet Dog costs 40 cents a can and has 20 units of a vitamin complex; the calorie content is 75 calories. Chow Hound cost 32 cents a can and has 35 units of vitamins and 50 calories. Kevin likes Amadeus to have at least 1175 units of vitamins a month and at least 2375 calories during the same time period. Kevin has space to store only 60 cans of dog food at a time. How much of each kind of dog food should Kevin buy each month to minimize his cost?

for this problem i made an equation and i set up a graph and matrix. but i cant seem to figure out condense the formula.

2) (a). The 11^th term of an A.P is -17 and the 32^nd term is -59
(i). The first term and the common difference.
(ii). The 16^th and the 26^th terms
(iii). The sum of the first 10 terms

(b) proofs.
(i). If the n^th term and the sum of n terms of an A.P are p and q respectively, prove that its first term is (2q-pn)/n


for this one i got -3 as the first term and -2 and the common difference, but that does not seem to be right, and for part b, i honestly have no idea.

 

[ksdhart]

To give good hints for your first problem, we'd need to see your work and your reasoning. You say you formed an equation and set up a graph and a matrix. So, when you reply back please tell us what equation(s) did you form? What graph did you draw? And what matrix did you set up? Thank you.

As for the problem 2a, you're on the right track. You can check your answer by plugging in the values. You say the common difference is -2 and the first term is -3. Okay, so then work with that series. The second term is then -5, the third -7, etc. But, then the 11th term is -23. But that doesn't match what you were given, so something's wrong. Let's go back to the drawing board then...

You're given the 11th and 32nd terms of the sequence. What's the difference between those two terms? And how many terms are there between them? Divide those two and you have your common difference. Once you have that, work backwards from the 11th term. To get to the 1st term, you'd need to go back 10 terms. You know the common difference, so subtract 10 times the common difference and you have your first term. From there, parts ii and iii are easily derived.

Problem 2b is certainly intimidating, but if you break it down, it's not quite as bad as it seems. You're told that the sum of the first n numbers in a sequence is some value q and the nth term is some other value p. The question then is, how do you calculate the sum of the first n numbers and how you calculate the nth term? To make it simple, I'll use a sequence that starts with 1 and adds 3 each time. Make a table of n values, and note the sums as you go along...

n=1; 1st term is 1; Sum is 1
n=2; 2nd term is (1+3) = 4; Sum is 1+(1+3) = 5
n=3; 3rd term is (1+3+3) = 7; Sum is 1+(1+3)+(1+3+3) = 12
n=4; 4th term is ?; Sum is ?

Can you derive a general formula for the sum, and another formula for the nth term? With a bit of simplification, I bet a formula for the first term can also be found. Good luck - it is quite a problem.