Question

Sum of first hundred numbers common to the two A.P.’s 12, 15, 18, … and 17, 21, 25 …..is​

Answer 1

Topic - Arithmetic Progression

First A.P.:

• 12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45, ...

Second A.P.:

• 17, 21, 25, 29, 33, 37, 41, 45, ...

We have to find the sum of the first 100 numbers common to the two A.P.'s. and we form the A.P. of common terms:

21, 33, 45, ...

• First term, a = 21
• Common difference, d = 12
• Number of terms, n = 100

∴ the sum of the first 100 terms is

= n/2 * [ 2a + (n - 1) d ]

= 100/2 * [ 2 * 21 + (100 - 1) * 12 ]

= 50 * [ 42 + 99 * 12 ]

= 50 * [ 42 + 1188 ]

= 50 * 1230

= 61500.

Answer 2

The sum of first hundred numbers common to the two arithmetic progression is 61500.

Step-by-step explanation:

First series:

12, 15, 18, 21, 24, 27, 30, 33, 36, 39, 42, 45,....

Second series:

17, 21, 25, 29, 33, 37, 41, 45, ...

The common numbers in the given series are:

21, 33, 45, ...

The sum of numbers are given by the formula:

S = n/2 [2a + (n - 1) d]

Where,

n = 100

a = 21

d = 33 - 21 = 12

On substituting the values, we get,

S = 100/2 [(2 × 21) + (100 - 1)12]

S = 50 [42 + (99 × 12)]

S = 50 [42 + 1188]

S = 50 × 1230

∴ S = 61500