1.
Which represents the first two terms of the sequence:
a_1 = 2 and a_n = -2(a_n-1)^2
option a. -8, -128
option b. 16, 1024
option c. -2, 16
option d. 2, -8
2.
Which is the seventh term in the sequence:
a_n = -1/125 * 5^n-1
option a. -125
option b. -625
option c. -25
option d. -5
3.
Which is the 10th term in the sequence:
-62, -47, -32, -17, -2, ...
option a. 58
option b. 88
option c. 73
option d. 12
4.
Which is the 14th term of the sequence:
1/81, 1/27, 1/9, 1/3, 1,...
option a. 2187
option b. 6561
option c. 19,683
option d. 59,049
#1,2 - just plug in a value for n
#3 Tn = -62 + 15(n-1) = -77+15n
#4 Tn = 1/243 * 3^n
1. To find the first two terms of the sequence defined by a_1 = 2 and a_n = -2(a_n-1)^2, we can use the recursive formula to find a_2.
Given that a_1 = 2, we can substitute n = 2 into the recursive formula:
a_2 = -2(a_2-1)^2
Substituting a_2-1 with a_1, we have:
a_2 = -2(a_1)^2 = -2(2)^2 = -2(4) = -8
Therefore, the first two terms of the sequence are -8 and 2. Therefore, option d) 2, -8 is the correct answer.
2. The formula provided is a geometric sequence with a common ratio of -1/125. To find the seventh term, we substitute n = 7 into the formula:
a_7 = -1/125 * 5^(7-1)
= -1/125 * 5^6
Simplifying:
a_7 = -1/125 * 15625
= -15625/125
= -125
Therefore, the seventh term of the sequence is -125. Therefore, option a) -125 is the correct answer.
3. The given sequence appears to be an arithmetic sequence with a common difference of 15. To find the 10th term, we can use the formula for arithmetic sequences:
a_n = a_1 + (n-1)d
Plugging in the known values:
a_10 = -62 + (10-1)*15
= -62 + 9*15
= -62 + 135
= 73
Therefore, the 10th term of the sequence is 73. Therefore, option c) 73 is the correct answer.
4. The given sequence is a geometric sequence with a common ratio of 1/3. To find the 14th term, we substitute n = 14 into the formula:
a_14 = (1/81) * (1/3)^(14-1)
= (1/81) * (1/3)^13
= (1/81) * (1/3^13)
= (1/81) * (1/1594323)
= 1/129140163
Therefore, the 14th term of the sequence is 1/129140163. None of the provided options match this value.
1. To find the first two terms of the sequence, we can use the recursive definition provided: a_n = -2(a_n-1)^2. We know that a_1 = 2, so we can use this information to find a_2.
a_2 = -2(a_1)^2
a_2 = -2(2)^2
a_2 = -2(4)
a_2 = -8
So the first two terms of the sequence are 2 and -8.
Now, let's check the options:
a. -8, -128: These are the correct first two terms.
b. 16, 1024: These are not the first two terms of the sequence.
c. -2, 16: These are not the first two terms of the sequence.
d. 2, -8: These are not the first two terms of the sequence.
Therefore, the correct answer is option a. -8, -128.
2. In the given sequence, a_n = -1/125 * 5^(n-1). We need to find the seventh term, a_7.
a_7 = -1/125 * 5^(7-1)
a_7 = -1/125 * 5^6
a_7 = -1/125 * 15625
a_7 = -125
So the seventh term of the sequence is -125.
Now, let's check the options:
a. -125: This is the correct seventh term.
b. -625: This is not the seventh term of the sequence.
c. -25: This is not the seventh term of the sequence.
d. -5: This is not the seventh term of the sequence.
Therefore, the correct answer is option a. -125.
3. The given sequence is -62, -47, -32, -17, -2, ... . It appears to be an arithmetic sequence with a common difference of 15.
To find the 10th term, we can use the formula for the nth term of an arithmetic sequence:
a_n = a_1 + (n-1)d
where a_n is the nth term, a_1 is the first term, n is the position of the term in the sequence, and d is the common difference.
Plugging in the values, we get:
a_10 = -62 + (10-1)(15)
a_10 = -62 + 9(15)
a_10 = -62 + 135
a_10 = 73
So the 10th term of the sequence is 73.
Now, let's check the options:
a. 58: This is not the 10th term of the sequence.
b. 88: This is not the 10th term of the sequence.
c. 73: This is the correct 10th term.
d. 12: This is not the 10th term of the sequence.
Therefore, the correct answer is option c. 73.
4. The given sequence is 1/81, 1/27, 1/9, 1/3, 1, ... . It appears to be a geometric sequence with a common ratio of 1/3.
To find the 14th term, we can use the formula for the nth term of a geometric sequence:
a_n = a_1 * r^(n-1)
where a_n is the nth term, a_1 is the first term, n is the position of the term in the sequence, and r is the common ratio.
Plugging in the values, we get:
a_14 = (1/81) * (1/3)^(14-1)
a_14 = (1/81) * (1/3)^13
a_14 = (1/81) * (1/3^13)
a_14 = (1/81) * (1/1594323)
a_14 ≈ 1/24,186,777
None of the given options match the result, so none of the options is correct.
Therefore, the answer is none of the options.