Sequences
1)A sequence has a first term of 24 and every other term is one half of the previous term.
a)Write the first four terms of the sequence.
b)Find a recursive formula that defines this sequence.
c)Find an explicit formula that defines this sequence. This will be discussed later, but you could use the geometric form tn=ar(n-1) where a is the first term, r is the common ratio and n is the number of terms.
d)Find t6 and t8.
Either you are lost, or are looking for someone to give you answers. I will assume the former for now:
a) 24, 12, 6,3, ...
b) an=an-1*1/2
c. tn=24*(1/2)n
t6= 24*(1/2)^6=24/64
I am looking for help to find out if my answer is right or not!
Thanks any way!!
To answer the questions about the given sequence, we will go through each part step by step:
a) To write the first four terms of the sequence, we start with the given first term, which is 24. Then, to find the next term, we divide the previous term by 2. So:
First term: 24
Second term: 24 / 2 = 12
Third term: 12 / 2 = 6
Fourth term: 6 / 2 = 3
Therefore, the first four terms of the sequence are 24, 12, 6, and 3.
b) To find the recursive formula that defines this sequence, we need to express each term in terms of the previous term(s). In this case, every other term is one half of the previous term. So, we can write the recursive formula as:
t(n) = (1/2) * t(n-1)
In this formula, t(n) represents the nth term of the sequence, and t(n-1) represents the previous term.
c) To find an explicit formula that defines this sequence, we can use the geometric form tn = a * r^(n-1), as mentioned in the question. Here, a is the first term, r is the common ratio, and n is the number of terms.
For this sequence, a (the first term) is 24, and the common ratio (r) is 1/2 (since each term is half of the previous term). Therefore, the explicit formula for this sequence is:
tn = 24 * (1/2)^(n-1)
d) To find t6 and t8, we substitute the respective values of n into the explicit formula:
For t6:
tn = 24 * (1/2)^(6-1)
tn = 24 * (1/2)^5
tn = 24 * (1/32)
tn = 3/2
Therefore, t6 = 3/2.
For t8:
tn = 24 * (1/2)^(8-1)
tn = 24 * (1/2)^7
tn = 24 * (1/128)
tn = 3/8
Therefore, t8 = 3/8.
So, t6 = 3/2 and t8 = 3/8 are the respective values of the sixth and eighth terms in the sequence.