Question

hey help me with this one (S1,S2,S3,......Sk) is a sequence where S1 is the sum of infinite numbers of a geometric sequence whose first term is 1 and ratio =2/3, S2 is the sum of infinite numbers of geometric sequence whose first term 2 and ratio=3/5 , S3 is sum of inifinite numbers of geometric sequence whose first term is 3 and ratio =4/7 and Sk is the sum to inifinity of geometric sequence whose first term is =k and ratio=k+1/2k+1
evaluate (S1+S2+S3+...........Sk)

Answer 1

S1 = 1/(1-2/3) = 1/(1/3) = 3
S2 = 2*[1/(1-3/5)]= 2(1/(2/5)) = 2*(5/2) = 5
S3 = 3*[1/(3/7)] = 3*(7/3) = 7
S4 = 4* [1/(4/9)] = 4(9/4) = 9
etc
So the sequence just becomes an arithmetic sequence with initial term 3 and common difference 2. The sum of the first k terms of an arithmetic sequence is given by (s1+sk)*k/2

Answer 2

so what is the number of terms in that arithmetic sequence

Answer 3

Sk = k*(1/(1-(k+1/2k+1)) = k*(1/(k/2k+1)) = k*((2k+1)/k) = 2k+1
so Sk = 2k+1
This shows that the sequence is arithmetic with initial value 3 and common difference 2.

Answer 4

ok i got that result too 2k+1 and i got that n=k

Answer 5

but still cant figure out what k or n is

Answer 6

Well, I mean, you could list an infinite number of terms. They're not asking you to sum any specific number of terms. They're just asking for the formula for the sum of the first k terms, k being an arbitrary variable.

Answer 7

n is not a variable that is mentioned in the problem. Did you define that for yourself?

Answer 8

And k is not meant to be solved for. It's meant to remain a variable.

Answer 9

does it really has no numerical value just formula

Answer 10

Yes. If they ask you about a specific number of terms, then you can substitute that number for k to figure out the sum.