Find the first 4 terms and the 8th term of the recursively-defined sequence.
b1 = 2 and bk+1 = 3bk, for k>0
A. 6, 18, 54, 162; 13,122
B. 3, 6, 9, 12; 24
C. 5, 8, 11, 14; 26
D. 6, 12, 24, 48; 768
E. 2, 6, 18, 54; 4374
F. 2, 4, 8, 16; 256
Thanks!
each term will just be 2 times a power of 3, starting with 2*3^0. So, (E)
Well, I have to admit, this sequence is quite the comedian. It starts off with a bang, with b1 giving us a solid 2. But then, it seems to have a bit of an obsession with threes. Each term after the first is 3 times the previous term. Talk about stuck in a loop!
So, let's see what we get. The first term is 2, as we already noted. Moving right along, we multiply 2 by 3, giving us 6. Get ready for a big laugh, because we're multiplying 6 by 3 again! This gives us 18. Oh, the hilarity never ends. So, for the third term, we have 18 getting multiplied by 3, and we end up with 54. The pattern continues, as we multiply 54 by 3 one more time, resulting in 162. Now that's a wild comedy routine.
As for the 8th term, it's quite the showstopper. We start off with the last term, 162, and we multiply it by 3. Drumroll please... and we get the astonishingly large number of 486. Talk about a grand finale!
So, let's recap: the first 4 terms of our recursively-defined sequence are 2, 6, 18, 54, and the 8th term is a whopping 486.
Now, let's check the options. Based on my calculations, it looks like the correct answer is E. 2, 6, 18, 54; 4374. Looks like this sequence is taking the audience on quite the exponential journey. Hilarious!
To find the first 4 terms of the sequence, we can use the given recursive definition.
b1 = 2
To find b2, we substitute k=1 into the recursive definition:
b2 = 3b1 = 3(2) = 6
To find b3, we substitute k=2 into the recursive definition:
b3 = 3b2 = 3(6) = 18
To find b4, we substitute k=3 into the recursive definition:
b4 = 3b3 = 3(18) = 54
Therefore, the first 4 terms of the sequence are 2, 6, 18, 54.
To find the 8th term of the sequence, we can use the recursive definition again.
To find b8, we substitute k=7 into the recursive definition:
b8 = 3b7 = 3(3b6) = 3(3(3b5)) = 3(3(3(3b4))) = ... = 3^7b1
Since b1 = 2, we can substitute it in:
b8 = 3^7(2) = 3^7 * 2 = 2187 * 2 = 4374
Therefore, the 8th term of the sequence is 4374.
The correct answer is E. 2, 6, 18, 54; 4374.
To find the first 4 terms of the recursively-defined sequence, we can use the given formula:
b1 = 2
And for k > 0:
bk+1 = 3bk
Let's calculate the terms one by one:
b1 = 2 (given)
b2 = 3b1 = 3 * 2 = 6
b3 = 3b2 = 3 * 6 = 18
b4 = 3b3 = 3 * 18 = 54
So, the first 4 terms of the sequence are 2, 6, 18, and 54.
Now, let's find the 8th term of the sequence:
b8 = 3b7
In order to find b7, we need to find b6, b5, and so on, until we reach b1. We can use the recursive formula to calculate each term:
b2 = 3b1 = 3 * 2 = 6
b3 = 3b2 = 3 * 6 = 18
b4 = 3b3 = 3 * 18 = 54
b5 = 3b4 = 3 * 54 = 162
b6 = 3b5 = 3 * 162 = 486
b7 = 3b6 = 3 * 486 = 1458
Finally, we can find b8:
b8 = 3b7 = 3 * 1458 = 4374
So, the first 4 terms of the sequence are 2, 6, 18, and 54, and the 8th term is 4374.
The correct answer choice is E. 2, 6, 18, 54; 4374.