RD Sharma Test: Arithmetic Progressions - Class 10 MCQ

Progressions with equal common difference are known as Arithmetic Progression.

∴ a_mn = a + (mn - 1) d = 1/mn + (mn - 1) 

Given: a₂₀ - a₁₃ and d = 5
⇒ a₂₀ - a₁₃ = a + (20 - 1) d - [a + (13 - 1) d] = a + (20 - 1) x 5 - [a + (13 - 1) x 5]
⇒ a₂₀ - a₁₃ - a = a + 95 - [a + 60]
= a+95 - a - 60 = 35 

The formula for n^th term of an AP is aₙ = a + (n - 1) d

Here, aₙ is the n^th term, a is the first term, d is the common difference and n is the number of terms.

For first A.P., a₁₀₀ = a₁ + (100 - 1) d

a₁₀₀ = a₁ + 99d ------ (1)

a₁₀₀₀ = a₁ + (1000 - 1)d

a₁₀₀₀ = a₁ + 999d ------ (2)

For second A.P.,

b₁₀₀ = b₁+ (100 - 1)d

b₁₀₀ = b₁ + 99d ------ (3)

b₁₀₀₀ = b₁ + (1000 - 1)d

b₁₀₀₀ = b₁ + 999d ------ (4)

Given that, difference between 100^th term of these A.P.s = 100

Thus, from equations (1) and (3) we have

(a₁ + 99d ) - (b₁ + 99d ) = 100

a₁ - b₁ = 100 ...(5)

Difference between 1000^th terms of these A.P.s

Thus, from equations (2) and (4) we have

(a₁ + 999d ) - (b₁ + 999d ) = a₁ - b₁

But a₁ - b₁ = 100 [From equation(5)]

Hence, the difference between the 1000^th terms of these A.P. will be 100.

Thus, initial term of the A.P. is 5 and the common difference is 6.

First Term =  a+b
Second Term =  a-b
Common Difference is a-b-a-b = -2b.

Summation of 22 terms of an A.P. is     
n/2 [ 2a + (n-1)d ]
22/2 [2 ( a+b) + (22-1)-2b ]
11  [ 2a+2a-b + (21)-2b  ]
11  [  2a+2b - 42b ]
11  [ 2a - 40b ]
22a - 40b.

So, the summation of given A.P. for 22 terms is 22a - 40b 

Let the middle term be a, then first term is a−d and next term is a+d

Odd numbers between 0 and 50 are 1, 3, 5, 7, ………, 49 Here a = 1,d = 3−1 = 2 and 

If a, b and c are in A.P., then 

Three digits numbers divisible by 7 are 105, 112, 119,.........., 994
Here, a= 105, d = 112 - 105 = 7, a_n = 994 

Let the three angles of a triangle be a - d, a and a + d.

Therefore, one angle is of 60° and other is 90° (given). Let third angle be x° . then

Therefore the angles of the right angled triangle are 30°, 60°, 90°.

Let the first prize be a.
The seven prizes form an AP with first term a and common difference d = −20
Now the sum of all seven prizes = Rs. 700