My ‘attempt’ is pretty disastrous but I haven’t found a better way to explain what I’m doing. Please bear with me.
Q. The 1st, 3rd and 6th terms of an arithmetic sequence are also in geometric sequence. Find the common ratio (r) of the geometric sequence.
A.: From text book, r = 3/ 2
Attempt:
If Un = a + (n - 1)d, assume d = 1, 2 and 3 respectively. This will result in the following sequences:
If d = 1: Un-1 = 1, 2, 3, 4, 5, 6
If d = 2: Un-1 = 2, 4, 6, 8, 10, 12
If d = 3: Un-1 = 3, 6, 9 12, 15, 18
From the question statement, we know that the 1st, 3rd and 6th numbers of the arithmetic sequence are in order for the geometric sequence.
In sequence Un-1, where d = 1
1, 3, 6 = U1, U2 and U3 of the geometric sequence respectively.
Similarly, where d = 2, we have:
2, 6, 12
And when d = 3:
3, 9, 18
Note in each case: U3/ U2 = 2 and U2/ U1 = 3, e.g. in the sequence where d = 3:
18/ 9 = 2 & 9/ 3 = 2
Therefore, r = 3/2, i.e (U2/ U1)/ (U3/ U2)…?
However,…
Geometric sequence: Un = ar^n-1
Going by geometric sequence layout, U1, U2, U3 = a, ar, ar^2, respectively.
Which means that in, say Un-1, d = 3, the sequence 3, 9, 18 should really be 3, 9, 27, i.e we say r = 3 and so: U1 = 3, U2 = 3.3, U3 = 3.3^2
Can anyone help me confirm if I‘m right, or completely off the mark? Thank you.
r is not (U3/U2)/(U2/U1),but it is U3/U2=U2/U1,and only when these two are equal.18/ 9 = 2 & 9/ 3 = 2
Therefore, r = 3/2
so none of the cases you mentioned satisfy the propositions.
the answer:
we have:
U3=U2+d=(U1+d)+d=U1+2d
and:
U6=U5+d=...=(U1+4d)+d=U1+5d
so (U1+5d)/(U1+2d)=(U1+2d)/U1
because both are equal to r
(U1+2d)(U1+2d)=U1(U1+5d)
U1^2+4dU1+4d^2=U1^2+5dU1
subtract U1 from both sides:
4dU1+4d^2=5dU1
4d^2=dU1
divide both by d
4d=U1
next:
r=U3/U1
r=(U1+2d)/U1
r=(4d+2d)/4d
r=6d/4d
r=6/4
r=3/2
hope it helps!
Ok, I'm starting to understand this but, Anonimnystefy, at the end of your explanation...
How did you arrive at U3/ U1? Is it not meant to be U2/U1?next:
r = U3/U1
And Plato, when you say:
How did you get ar + 2dr = a +5d? Is it not ar^2 = a+ 5d?
I don't mean to come across like an ass, but if I understand how you worked those parts out, I should be able to piece this together. Thank you for all the help so far.