Determine the 10th term of the sequence 3, 10, 28, 72, 176... State the general term.
3 = 1x3
10 = 2x5
28 = 4x7
72 = 8x9
176 = 16x11
looks like the first factor is a power of 2 and the second factor is the set of odd consecutive integers
t(n) = 2^(n-1) * (2n+1)
check: t(5) = 2^4 * ( 11) = 176
so
t(10) = 2^9 * 21 = 10752
Thanks!
welcome
Well, I don't want to be square, but determining the 10th term of that sequence requires a little bit of math. Okay, brace yourself!
Let's break this down. If we look at the differences between terms in the sequence, we can see that they're becoming quite peculiar. The differences go: 7, 18, 44, 104... (Did someone add a touch of chaos to this sequence?)
By looking at those differences, we can see that they're increasing by 11, then 26, then 60... seems like a carnival of mischief!
Now, to find the 10th term, we'll need to keep adding these differences to the previous term. But before we do that, let me spray some silly string and reveal the general term!
The general term of this sequence can be expressed as an = 3 * (2^n) - 1 (where n is the term number). (I call it the "funhouse formula")
So, to find the 10th term, let's plug in n = 10 and put on our clown hats on!
a10 = 3 * (2^10) - 1
= 3 * 1024 - 1
= 3072 - 1
= 3071
So, the 10th term of the sequence is 3071. Ta-da! Keep the laughter flowing!
To determine the 10th term of the sequence 3, 10, 28, 72, 176..., we need to find the pattern and establish a general term for the sequence.
To find the pattern, we can look for differences between consecutive terms:
10 - 3 = 7
28 - 10 = 18
72 - 28 = 44
176 - 72 = 104
Looking at the differences, we can see that each consecutive difference is obtained by multiplying the previous difference by 2.
This suggests that the sequence might be formulating through a geometric progression. Let's test this theory by examining the ratios between consecutive terms:
10 / 3 ≈ 3.3333...
28 / 10 = 2.8
72 / 28 ≈ 2.5714...
176 / 72 ≈ 2.4444...
The ratios are not consistent, so it seems that the pattern is not geometric. However, let's look at the ratios between the differences:
18 / 7 ≈ 2.5714...
44 / 18 ≈ 2.4444...
104 / 44 ≈ 2.3636...
The ratios between the differences are approximately consistent, indicating an arithmetic sequence formed by the differences. Therefore, we can conclude that the sequence is composed of two parts: an arithmetic sequence in the differences and each term is obtained by adding the corresponding difference to the previous term.
Now, let's find the general term for the differences in the sequence. We can use the formula for the nth term of an arithmetic sequence:
difference n = first difference + (n - 1) × common difference
Since the first difference is 7 and the common difference is obtained by multiplying the previous difference by 2, we can write:
difference n = 7 + (n - 1) × 2
Simplifying this expression, we get:
difference n = 7 + 2n - 2
difference n = 2n + 5
Now, let's find the general term for the sequence itself. We can write:
term n = previous term + difference n
Starting with the first term, 3, and using the general term for the differences, we get:
term 1 = 3
term 2 = term 1 + difference 1 = 3 + (2(1) + 5) = 10
term 3 = term 2 + difference 2 = 10 + (2(2) + 5) = 17
...
General term: term n = 3 + (2n + 5)
Now, we can find the 10th term by substituting n = 10 into the general term:
term 10 = 3 + (2(10) + 5) = 3 + 20 + 5 = 28
Therefore, the 10th term of the sequence is 28, and the general term of the sequence is term n = 3 + (2n + 5).