14. Orlando is making a design for a logo. He begins with a square measuring 20 inches on a side. The second square has a side length of 16 inches, and the third square has a side length of
12.8 inches. Which square will be the first square with a side length of less than 8 inches?
A)seventh square
B) fourth square
C) fifth square
D) sixth square
15. What is the 50th term of the sequence that begins -2, 4, 10, 16
290
294
292
296
16. What is the 20th term of the sequence that begins -3, 6, -12, 24
-524,288
524, 288
1,572, 864
3,145,728
17. The 5th term ina geometric sequence is 160. The 7th is 40. Whar are the possible value of the 6th term of the sequence?
±70
70
±80
80
18. Find the sum of the following series
15∑n=1 (2n+1)
240
255
210
510
Answer:
14. D) sixth square.
To find the side length of the sixth square:
Let x be the side length of the sixth square.
The side length of each square is obtained by multiplying the side length of the previous square by 0.8.
So, we can write:
20 x 0.8^5 = x
x = 7.8125
Therefore, the sixth square has a side length of approximately 7.81 inches, which is less than 8 inches.
15. 296
The sequence is increasing by 6, 6, and 6 for each term. So, the 50th term would be:
-2 + (50-1)6 = 296
16. 1,572,864
The sequence is doubling for each term but with alternating signs. So, the 20th term would be:
-3 x 2^(20-1) = -3 x 524,288 = -1,572,864
17. ±80
We can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n-1), where a1 is the first term and r is the common ratio.
From the given information, we can write:
a5 = 160 = a1 * r^(5-1)
a7 = 40 = a1 * r^(7-1)
Dividing the second equation by the first, we get:
40/160 = (a1 * r^(7-1))/(a1 * r^(5-1))
1/4 = r^2
r = ±√(1/4) = ±1/2
Substituting this value of r in the first equation, we get:
160 = a1 * (1/2)^4
a1 = 160 x 16 = 2560
So, the sixth term would be:
a6 = a1 * r^(6-1) = 2560 x (1/2)^5 = 80 or -80
18. 510
To find the sum of the series:
15∑n=1 (2n+1) = 15(1 + 3 + 5 + ... + 39)
This is an arithmetic series with a common difference of 2.
The formula for the sum of an arithmetic series is:
Sn = n/2(a1 + an), where n is the number of terms, a1 is the first term, and an is the nth term.
In this case, we have:
a1 = 1
an = 39
n = (an - a1)/d + 1, where d is the common difference.
n = (39 - 1)/2 + 1 = 20
So, the sum of the series is:
S20 = 20/2(1 + 39) = 510.
14. To find the first square with a side length of less than 8 inches, we can find the side lengths of the squares and compare them to 8 inches.
The side lengths of the squares are as follows:
First square: 20 inches
Second square: 16 inches
Third square: 12.8 inches
Comparing the side lengths to 8 inches, we can see that the third square with a side length of 12.8 inches is the first square with a side length less than 8 inches.
Therefore, the answer is C) fifth square.
15. The given sequence begins with -2, 4, 10, 16. We can observe that each term in the sequence is obtained by adding 6 to the previous term.
To find the 50th term, we can start with the first term (-2) and continue adding 6 repeatedly:
-2 + 6 = 4
4 + 6 = 10
10 + 6 = 16
...
Continuing this pattern, we can find the 50th term:
-2 + (49 * 6) = -2 + 294 = 292
Therefore, the 50th term of the sequence is 292. The answer is C) 292.
16. The given sequence begins with -3, 6, -12, 24. We can observe that each term alternates between being positive and negative, and is obtained by multiplying the previous term by -2.
To find the 20th term, we can start with the first term (-3) and continue multiplying by -2 repeatedly:
-3 * -2 = 6
6 * -2 = -12
...
Continuing this pattern, we can find the 20th term by multiplying it -2 a total of 9 times:
-3 * (-2)^9 = -3 * 512 = -1,536
Therefore, the 20th term of the sequence is -1,536. The answer is not listed, as the correct answer would be -1,536.
17. In a geometric sequence, each term is obtained by multiplying the previous term by a common ratio. Let's denote the common ratio as r.
From the given information, we know that the 5th term is 160 and the 7th term is 40. We can use these values to find the common ratio.
160 / r^2 = 40
Dividing both sides by 40:
4 / r^2 = 1
Multiplying both sides by r^2:
4 = r^2
Taking the square root of both sides:
r = ±2
Now, we can find the 6th term by multiplying the 5th term by the common ratio:
160 * 2 = 320
160 * (-2) = -320
Therefore, the possible values for the 6th term are ±320. The answer is not listed, as the correct answer would be ±320.
18. To find the sum of the given series, we can use the formula for the sum of an arithmetic series:
Sn = n/2 * (a + l)
where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, the given series is:
15 * Σn=1 (2n+1)
To simplify, we can expand the sum:
15 * (2(1) + 1 + 2(2) + 1 + 2(3) + 1 + ... + 2(15) + 1)
Simplifying further, we can separate the constants and rewrite the sum:
15 * [(2(1) + 2(2) + 2(3) + ... + 2(15)) + (1 + 1 + 1 + ... + 1)]
The first part of the sum is a arithmetic series with the first term 2, common difference 2, and the number of terms 15:
15 * (n/2 * (2 + (2 * (15-1))))
= 15 * (15/2 * (2 + 28))
= 15 * (15/2 * 30)
= 15 * (225)
= 3375
The second part of the sum has 15 terms, each being 1:
15 * 1 = 15
Adding the two parts together:
3375 + 15 = 3390
Therefore, the sum of the series is equal to 3390. The answer is not listed, as the correct answer would be 3390.