Quiz On Mathematics 10

Explanation
To find the sum of all the odd integers between 8 and 26, we need to identify the odd numbers in that range and then add them together. The odd numbers between 8 and 26 are 9, 11, 13, 15, 17, 19, 21, 23, and 25. Adding these numbers together, we get a sum of 153.

Explanation
The given geometric sequence is formed by multiplying each term by -3. Therefore, to find the next term, we need to multiply the last term, 36, by -3. This gives us -108, which is the next term in the sequence.

Explanation
The given sequence is an arithmetic progression with a common difference of 7. To find the second arithmetic mean, we need to find the number that comes after the first arithmetic mean. Since the first arithmetic mean is 18, we can add the common difference of 7 to find the second arithmetic mean, which is 25.

Explanation
The question asks for the third geometric mean between 1 and 256. To find the geometric mean, we need to take the square root of the product of the two numbers. In this case, the first number is 1 and the second number is 256. Taking the square root of 256 gives us 16, which is the first geometric mean. To find the second geometric mean, we take the square root of the product of 1 and 16, which is 4. Finally, to find the third geometric mean, we take the square root of the product of 1 and 4, which is 2. However, the given answer of 64 does not match this pattern and seems to be incorrect.

Explanation
The given arithmetic sequence starts with 4 and has a common difference of -3. To find the term that is -29, we can use the formula for the nth term of an arithmetic sequence: an = a1 + (n-1)d. Plugging in the values, we get a12 = 4 + (12-1)(-3) = 4 + 11*(-3) = 4 - 33 = -29. Therefore, the -29th term is the 12th term of the sequence.

Explanation
In an arithmetic sequence, each term is obtained by adding a constant value (called the common difference) to the previous term. To find the common difference, we can subtract the first term (2) from the 18th term (87), giving us a difference of 85. Since there are 17 terms between the first and 18th term, we can divide the difference by 17 to find the common difference. 85 divided by 17 equals 5. Therefore, the common difference of the sequence is 5.

Explanation
The Fibonacci sequence is a series of numbers in which each number is the sum of the two preceding ones. In this sequence, the first two terms are 1 and 1. To find the next term, we add the previous two terms: 1 + 1 = 2. Continuing this pattern, the next term would be 3 (2 + 1). Therefore, the correct answer is 3, not 13.

Explanation
To find the sum of a geometric sequence, we can use the formula S = a * (1 - r^n) / (1 - r), where S is the sum, a is the first term, r is the common ratio, and n is the number of terms. In this case, the first term is 3, the common ratio is 5, and the last term is 46,875. We can find n by solving the equation a * r^(n-1) = 46,875. Plugging in the values, we get 3 * 5^(n-1) = 46,875. Simplifying, we find that n-1 = 7, so n = 8. Plugging in the values into the sum formula, we get S = 3 * (1 - 5^8) / (1 - 5), which simplifies to 58,593.

Explanation
In a geometric sequence, each term is found by multiplying the previous term by a constant called the common ratio. To find the eighth term, we can use the formula: nth term = a * r^(n-1), where a is the first term, r is the common ratio, and n is the term number. Given that the third term is 27 and the common ratio is 3, we can solve for the first term using the formula: 27 = a * 3^(3-1). Simplifying this equation, we get: 27 = a * 9. Dividing both sides by 9, we find that the first term is 3. Now, substituting the values into the formula, we find the eighth term: 8th term = 3 * 3^(8-1) = 3 * 3^7 = 3 * 2187 = 6561.

Explanation
To find the sum of all the multiples of 3 from 15 to 49, we need to first find the first and last multiples of 3 in this range. The first multiple of 3 is 15, and the last multiple of 3 is 48. We can then use the formula for the sum of an arithmetic series, which is (n/2)(first term + last term), where n is the number of terms. In this case, n = (48-15)/3 + 1 = 12. Plugging in the values, we get (12/2)(15 + 48) = 6(63) = 378. Therefore, the correct answer is 378.

Explanation
The given arithmetic sequence starts with 7 and increases by 2 each time. This means that the common difference between consecutive terms is 2. The nth term of an arithmetic sequence can be found using the formula: nth term = first term + (n - 1) * common difference. In this case, the first term is 7 and the common difference is 2. Plugging these values into the formula, we get the expression 2n + 5, which represents the nth term of the sequence.

Explanation
To find the value of p that will make the numbers form an arithmetic sequence, we need to determine the common difference between the terms. In an arithmetic sequence, the common difference between consecutive terms remains constant.

By comparing the given terms, we can see that the common difference is the same between each pair of consecutive terms.

The common difference between the first and second terms is (5p+12) - (7p+2) = -2p + 10.
The common difference between the second and third terms is (2p-1) - (5p+12) = -3p - 13.

Since these two common differences are equal, we can set them equal to each other and solve for p:

-2p + 10 = -3p - 13
p = -23

Therefore, the value of p that will make the numbers form an arithmetic sequence is -23.

Explanation
The numbers 2k + 1, 3k + 4, and 7k + 6 form a geometric sequence if the ratio between any two consecutive terms is constant. To find this constant ratio, we can divide the second term by the first term and the third term by the second term.

(3k + 4) / (2k + 1) = (7k + 6) / (3k + 4)

Simplifying this equation, we get:

(3k + 4)^2 = (2k + 1)(7k + 6)

Expanding and simplifying further, we get:

9k^2 + 24k + 16 = 14k^2 + 13k + 6

Rearranging and simplifying, we get:

5k^2 - 11k + 10 = 0

Factoring this quadratic equation, we get:

(k - 2)(5k - 5) = 0

Solving for k, we get two possible values: k = 2 or k = 1. However, substituting k = 1 into the original equation does not satisfy the condition for a geometric sequence. Therefore, the correct value for k is 2. The answer is 2; -1.

Explanation
The car's value depreciates by 10% each year. After the first year, the car's value is 90% of Php 600,000, which is Php 540,000. After the second year, the car's value is 90% of Php 540,000, which is Php 486,000. After the third year, the car's value is 90% of Php 486,000, which is Php 437,400. After the fourth year, the car's value is 90% of Php 437,400, which is Php 393,660. Therefore, the value of the car after 4 years is Php 393,660.

Explanation
The skydiver jumps 16 feet on the first fall, 48 feet on the second fall, and 80 feet on the third fall. We can observe that the distance jumped increases by 32 feet with each subsequent fall (48 - 16 = 32 and 80 - 48 = 32). Therefore, for the tenth fall, the skydiver will jump an additional 32 feet compared to the ninth fall. Since the skydiver jumped 80 feet on the ninth fall, he will jump 80 + 32 = 112 feet on the tenth fall.

Explanation
Carl decided to give Nicole flowers according to the Fibonacci sequence. The Fibonacci sequence starts with 1 and 2, and each subsequent number is the sum of the two preceding numbers. So, on the first day, Nicole received 1 rose, on the second day she received 2 roses, on the third day she received 3 roses, on the fourth day she received 5 roses, and so on. To find out how many roses Nicole received on the tenth day, we continue the sequence until the tenth day. The sequence would be: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. Therefore, Nicole received 89 roses on the tenth day.

Explanation
The process of joining the midpoints of consecutive sides of a square creates a smaller square that is half the size of the original square. Since there are six squares formed, the sizes of the squares would be 8 inches, 4 inches, 2 inches, 1 inch, 0.5 inches, and 0.25 inches. The sum of the areas of these squares can be calculated by adding the areas of each square: 8^2 + 4^2 + 2^2 + 1^2 + 0.5^2 + 0.25^2 = 64 + 16 + 4 + 1 + 0.25 + 0.0625 = 85.3125 square inches, which is closest to 126.

Explanation
In this scenario, the text messages were sent in a branching pattern. The principal sends the first message, which is received by two teachers. Each of these two teachers then sends the message to two other teachers, resulting in a total of four messages sent in the second round. In the third round, each of the four teachers sends the message to two more teachers, resulting in a total of eight messages. This pattern continues for five rounds. So, in the fifth round, there would be 2^5 = 32 messages sent. However, since the principal's message is already counted as the first message, the total number of text messages sent would be 31.

Explanation
A geometric sequence is a sequence where each term after the first is obtained by multiplying the preceding term by the same constant. This means that each term in the sequence is a multiple of the previous term, resulting in a pattern of exponential growth or decay. In contrast, a harmonic sequence is a sequence where each term is the reciprocal of a corresponding term in an arithmetic sequence. An infinite sequence is a sequence that continues indefinitely, while a finite sequence has a limited number of terms. Therefore, the given explanation aligns with the concept of a geometric sequence.

Explanation
A harmonic sequence is a sequence in which the reciprocals of the terms form an arithmetic sequence. In other words, the difference between consecutive reciprocals is constant. This means that each term in the sequence is the reciprocal of its corresponding term in the arithmetic sequence. Therefore, the given answer, "Harmonic sequence," is correct.