1. Find the general rule for each arithmetic or geometric sequence.
2. Calculate the 10th term for each sequence.
a) a=a sub n-1*n
a sub one=9
b) a sub n=a sub n-1 + 8
a sub 1=-17
c) a sub n=a sub n-1 + 2
a sub 1=9
d) a=n*a sub n-1
a sub one=1
I don't understand how to determine if it is an arithmetic sequence or a geometric sequence if there is no sequence and I don't understand how to find the general rule.
clearly you have skipped the definitions.
arithmetic has terms which differ by the same amount
geometric has terms with a common ratio, one to the next.
a) has a ratio which changes: A(n) = A(n-1)*n
A(n)/A(n-1) = n, which is not constant
not GP or AP
b)That +8 is a dead giveaway. Each term is 8 more than the last, so it's an AP
c) same idea, AP
d) exactly the same as (a)
To determine if a sequence is arithmetic or geometric, you'll need to examine the relationship between consecutive terms in the sequence.
For an arithmetic sequence, the difference between consecutive terms is always the same. In other words, you add or subtract the same value to get from one term to the next.
For a geometric sequence, each term is obtained by multiplying the previous term by a constant value.
Let's go through each question and determine the type of sequence and find the general rule:
a) We're given a recursive formula a = a sub n-1 * n and a sub 1 = 9. To find the general rule, we need to rewrite the recursive formula in terms of n and a sub 1.
Starting with a sub 1 = 9, let's write out the first few terms to see if there is a pattern:
a sub 1 = 9
a sub 2 = a sub 1 * 2 = 9 * 2 = 18
a sub 3 = a sub 2 * 3 = 18 * 3 = 54
a sub 4 = a sub 3 * 4 = 54 * 4 = 216
It appears that each term is obtained by multiplying the previous term by n. So the general rule for this sequence is a sub n = a sub n-1 * n.
b) We're given a recursive formula a sub n = a sub n-1 + 8 and a sub 1 = -17. Let's rewrite the recursive formula in terms of n and a sub 1 by substituting the given values:
a sub 1 = -17
a sub 2 = a sub 1 + 8 = -17 + 8 = -9
a sub 3 = a sub 2 + 8 = -9 + 8 = -1
a sub 4 = a sub 3 + 8 = -1 + 8 = 7
The terms in this sequence are obtained by adding 8 to the previous term. So the general rule is a sub n = a sub n-1 + 8.
c) We're given a recursive formula a sub n = a sub n-1 + 2 and a sub 1 = 9. Following the same process as before, let's rewrite the recursive formula in terms of n and a sub 1:
a sub 1 = 9
a sub 2 = a sub 1 + 2 = 9 + 2 = 11
a sub 3 = a sub 2 + 2 = 11 + 2 = 13
a sub 4 = a sub 3 + 2 = 13 + 2 = 15
The terms in this sequence are obtained by adding 2 to the previous term. So the general rule is a sub n = a sub n-1 + 2.
d) We're given a recursive formula a = n * a sub n-1 and a sub 1 = 1. Substituting the values into the formula:
a sub 1 = 1
a sub 2 = 2 * a sub 1 = 2 * 1 = 2
a sub 3 = 3 * a sub 2 = 3 * 2 = 6
a sub 4 = 4 * a sub 3 = 4 * 6 = 24
The terms in this sequence are obtained by multiplying the previous term by n. So the general rule is a sub n = n * a sub n-1.
Now, to calculate the 10th term for each sequence, you can use the general rule along with the given first term:
a) For a sub n = a sub n-1 * n and a sub 1 = 9, plug in n = 10 and a sub 1 = 9 into the general rule:
a sub 10 = a sub 10-1 * 10 = a sub 9 * 10
You'll need to recursively calculate the terms until you reach the 10th term.
Repeat the above calculation for the remaining sequences using their respective general rules and given first terms.