Sequences and Series Part 1
2. Write a recursive definition for the sequence 8, 6, 4, 2, … (1 point)
a_1 = 8; a_n= a_(n–1) − 2
a_1 = 2; a_n= a_(n–1) + 6
a_1 = 8; a_n= a_(n–1) + 2
a_1 = 2; a_n= a_(n–1) + 8
Looks like each term is 2 less than its previous one.
That's the key to the formula
so we have to start with a(1) = 8
then a(n) = a(n-1) - 2 , n>1
the first one matches that.
The terms are increasing by 5, so I would expect to see something like
a(n) = 5n + ?? , (that rules out #3)
when n = 1, a(1) = 5 + ?? , but that is supposed to be -4
5 + what = -4
what = -9
so a(n) = 5n - 9
I don't see a -9, but what if we expand their expressions
1st: a(n) = -4 + 5(n-1)
= -4 + 5n - 5
= 5n -9
Yeahhh! that's it. So it is the first one
To write a recursive definition for a sequence, we need to define the first term and the rule to obtain the next term in terms of the previous term.
In this case, the given sequence is 8, 6, 4, 2, ...
Let's go through the options provided:
a_1 = 8; a_n = a_(n–1) − 2
This option states that the first term is 8, and to obtain the next term, we subtract 2 from the previous term. This is correct because if we start with 8 and subtract 2, we get 6. If we subtract 2 again, we get 4, and so on. So, this is a valid recursive definition for the sequence.
a_1 = 2; a_n = a_(n–1) + 6
This option states that the first term is 2, and to obtain the next term, we add 6 to the previous term. However, this does not match the given sequence because if we start with 2 and add 6, we get 8, not 6.
a_1 = 8; a_n = a_(n–1) + 2
This option states that the first term is 8, and to obtain the next term, we add 2 to the previous term. However, this does not match the given sequence because if we start with 8 and add 2, we get 10, not 6.
a_1 = 2; a_n = a_(n–1) + 8
This option states that the first term is 2, and to obtain the next term, we add 8 to the previous term. However, this does not match the given sequence because if we start with 2 and add 8, we get 10, not 6.
Therefore, the correct recursive definition for the sequence 8, 6, 4, 2, ... is:
a_1 = 8; a_n = a_(n–1) − 2.
ty, can you help with this one:
3. Write an explicit formula for the sequence –4, 1, 6, 11,… (1 point)
a_n = –4 + 5(n + 1)
a_n = –4 + 5(n – 1)
a_n = –4 + (n – 1)
a_n = –4 + 5n