Arithmetic Problems

Definition:

Arithmetic involves the study of quantity, especially as the result of combining numbers. In common usage, Arithmetic refers to the simpler properties when using the traditional operations of addition, subtraction, multiplication and division with smaller values of numbers.

1) Question: The arithmetic mean between two terms in an arithmetic sequence is 21. If one term is 13, then find the other term.

A ) 58
B ) 50
C ) 29
D ) 21

Steps to derive

1 (a+b)2
[The arithmetic mean of two numbers a and b.]

2 (13+b)2 = 21
[Substitute the values.]

3 13 + b = 42

4 b = 29

5 So, the second term is 29.



Hence the right answer is Option C

2) Question: If 6 arithmetic means are introduced between 1 and 15, then find their sum.

A ) 64
B ) 48
C ) 2
D ) 32

Steps to derive

1 Let x₁, x₂, x₃, x₄, x₅, x₆ be the six arithmetic means introduced between 1 and 15 respectively.

2 1, x₁, x₂, x₃, x₄, x₅, x₆, 15 are in arithmetic progression.

3 Sum of the 8 terms = 8 / 2[1 + 15]
[The sum of the n terms S_n = n2[a₁ + a_n] where n is the number of terms, a₁ is the first term and a_n is the last term.]

4 = 4 × 16 = 64

5 Therefore, the sum of the six arithmetic means = x₁ + x₂ + x₃ + x₄ + x₅ + x₆ = 64 - 16 = 48



Hence the right answer is Option B