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Multiple choice maths average arithmetic mean of ap introduction to averages means

The ratio of sum of n arithmetic means between two given numbers to that of single arithmetic mean between them id

  1. n : 1
  2. n$^2$ : 1

  3. 1 : 1
  4. $\sqrt{n}$ : 1
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The sum of n arithmetic means between a and b is n * (a+b)/2. The single arithmetic mean is (a+b)/2. The ratio is therefore n : 1.

Multiple choice maths average arithmetic mean of ap introduction to averages means

If $\cfrac {a^n+b^n}{a^{n-1}+b^{n-1}}$ is the AM between a and b, then the value of n is 

  1. 0

  2. 1

  3. -1

  4. none of these
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

The A.M. of a and b is $\dfrac{a+b}{2}$.


Therefore, $\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}$ will be the A.M. of a and b, if 

$\dfrac{a^{n+1}+b^{n+1}}{a^n+b^n}=\dfrac{a+b}{2}$

$\Rightarrow 2(a^{n+1}+b^{n+1})=(a^n+b^n)(a+b)$

$\Rightarrow 2a^{n+1}+2b^{n+1}=a^{n+1}+a^nb+b^na+b^{n+1}$

$\Rightarrow a^{n+1}+b^{n+1}=a^nb+b^na$

$\Rightarrow a^n(a-b)=b^n(a-b)$

$\Rightarrow a^n=b^n$

$\Rightarrow \dfrac{a^n}{b^n}=1$

$\Rightarrow (\dfrac{a}{b})^n=1$

$\Rightarrow (\dfrac{a}{b})^n=(\dfrac{a}{b})^0$

$\Rightarrow n=0$

Multiple choice maths average arithmetic mean of ap introduction to averages means

Find the average of the following set of scores $253,124,255,534,836,375,101,443,760$

  1. $427$
  2. $413$
  3. $141$
  4. $409$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

To find the average, sum all the given scores and divide by the total count of numbers. The sum of 253, 124, 255, 534, 836, 375, 101, 443, and 760 is 3681, and dividing by 9 yields 409.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The average age of $30$ girls. is $13\ yr$. The average of first $18$ girls is $15\ yr$. Find out the average age of remaining $12$ girls? 

  1. $12\ yr$
  2. $10\ yr$
  3. $16\ yr$
  4. $10.5\ yr$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

Multiply the total count by the overall average to find the total sum of ages, which is 30 * 13 = 390. Next, find the sum of the ages of the first 18 girls by multiplying 18 * 15 = 270. Subtracting this from the total gives 390 - 270 = 120 for the remaining 12 girls, leading to an average of 120 / 12 = 10 years.

Multiple choice maths average arithmetic mean of ap introduction to averages means

A student bought $4$ books for $Rs.120$ from one book shop and $6$ books for $Rs.150$ from another. The average price (in  rupees), he paid per book was:

  1. $27$
  2. $27.50$
  3. $135$
  4. $138$
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Total cost = 120 + 150 = 270. Total books = 4 + 6 = 10. Average price = 270 / 10 = 27.

Multiple choice maths average arithmetic mean of ap introduction to averages means

Let $(1-2x+3x^{2})^{10}=a _{0}+a _{1}x+a _{2}x^{2}+....+a _{n}x^{n},a _{n}\neq 0$, then the arithmetic mean of $a _{0},a _{1},a _{2},....a _{n}$ is

  1. $\dfrac{1024}{11}$
  2. $\dfrac{512}{7}$
  3. $\dfrac{512}{11}$
  4. $\dfrac{1024}{21}$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The arithmetic mean of coefficients a_0, a_1, ..., a_n is the sum of coefficients divided by the number of terms, which is (1/21) * sum. By substituting x = 1 and x = -1 into the polynomial expansion, we find the sum of all coefficients and the alternating sum, allowing us to find the total sum of coefficients and compute the mean as 1024/21.

Multiple choice maths average arithmetic mean of ap introduction to averages means

If $a,b,c,d,e,f$ are $A.M.s$ between $2$ and $12$ then $a+b+c+d+e+f$ is equal to

  1. $14$
  2. $42$
  3. $84$
  4. $None\ of\ these$
Reveal answer Fill a bubble to check yourself
B Correct answer
Explanation

When n arithmetic means are inserted between two numbers p and q, their sum is equal to n times the single arithmetic mean of the two numbers, or n * (p + q) / 2. Here there are 6 means between 2 and 12, so the sum is 6 * (2 + 12) / 2 = 6 * 7 = 42.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The average height of 25 boys is 1.4 m. When 5 boys leave the group, then the average height increases by 0.15 m. What is the average height of the 5 boys who leave?

  1. 0.8 m

  2. 0.9 m

  3. 0.95 m

  4. 1.05 m
Reveal answer Fill a bubble to check yourself
A Correct answer
Explanation

Initial total height = 25 * 1.4 = 35. New total height = 20 * 1.55 = 31. Total height of 5 boys = 35 - 31 = 4. Average = 4 / 5 = 0.8.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The average age of 36 student in a group is 14 years. When teacher's age is included to it, the average increases by one. What is the teacher's age in years?

  1. 31

  2. 36

  3. 51

  4. Cannot be determined

  5. None of these
Reveal answer Fill a bubble to check yourself
C Correct answer
Explanation

The total age of 36 students is 36 * 14 = 504. When the teacher is added, there are 37 people and the average becomes 15, giving a total sum of 37 * 15 = 555. The teacher's age is the difference between these two totals, which is 555 - 504 = 51.

Multiple choice maths average arithmetic mean of ap introduction to averages means

The sum of first $n$ natural numbers is given by the expression $(2n^{2}+3n)$. The mean of the given numbers is 

  1. $\left(\dfrac{2}{n}+3\right)$
  2. $\left(2+\dfrac{3}{n}\right)$
  3. $(2+3n)$
  4. $(2n+3)$
Reveal answer Fill a bubble to check yourself
D Correct answer
Explanation

The mean of the first n natural numbers is their sum divided by n. Given the sum is 2n^2 + 3n, dividing this expression by n gives (2n^2 + 3n) / n = 2n + 3.