The ratio of the fifth term to the twelfth term of a sequence in an arithmetic progression is 6/13. If each term of this sequence is positive, and the product of the first term and the third term is 32, find the sum of the first 100 terms of this sequence.
1 2 3 4 5 6 7 8 9 10 11 12 13 6 13 A1*A2 = 32
6 + 7(d) = 13 7d = 7, d = 1 in this case, but i believe its scaled
1 2 3 4 5 6 7 8 9 10 11 12 13 2 3 4 5 6 7 8 9 10 11 12 13 14 2*4N = 32 8N = 32 N=4 multiply it all by 4
8,12,16,20,24,28,32,36,40,44,48, .... is the sequesnce
to sum up all the terms we do, n(first+last)/2 right?
to test this out, 8,12,16,32 = 68 4(8+32) -------- = 2*40 = 80 .... maybe needs to be remembered better on that part 2
8,12,16, 20 = 58 4(8+20) ------- = 2(28) = 56 ... helps of i use the right numbers lol 2
so now all we gots to do is figure out the 100th number 1 2 3 4 5 6 7 8 8,12,16,20,24,28,32,36,40,44,48 An=8+4(n-1) A(1) = 8 + 4(0) = 8 A(2) = 8 + 4(1) = 12 A(3) = 8 + 4(2) = 16 .... A(100) = 8 + 4(99) = 404
100(8+404) ----------- = 50(412) 2
Thanks for the answer but I'm a bit confused here. When you get the sequence 2,3,4.. What did you do next? what is N and why is there the 8?
the clues given tell us that the 1st and 3rd terms multiply to give us a product of 32
2*4 = 8 8 does not equal 32 unless we scale it (multiply it by some (N)umber)
8N = 32 when N=4
so multiply the whole lot by 4 to work the problem thru; its an intermediary step
The ratio of the 5th and 12th is 6/13. 4/4(6/13) = 24/52 so it works there too If the product of the 1st and 3rd is 32 2 * 4 = 8 ; not 32 ... so multiply it by 4, in fact multiply the sequence by 4 to get the proper results that fit
Join our real-time social learning platform and learn together with your friends!