The sum of the 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the A.P.

By Ritesh|Updated : November 10th, 2022

The first three terms of the A.P. are -13, -8, and -3. Steps to find the first three terms of the A.P.:

Step 1: Determine the integers:

The nth term the AP (arithmetic progression) is given the for:

an = a + (n - 1)d

where; a = first term of AP

and, d = a common difference between AP

and, an = nth term of AP

We know that, an = a + (n - 1)d

Therefore,

  • 4th term is given by: a4 = a + (4 - 1) d = a + 3d
  • 8th term is given by: a8 = a + (8 - 1)d = a + 7d
  • 6th term is given by: a6 = a + (6 - 1)d = a + 5d
  • 10th term is given by: a10 = a + (10 - 1)d = a + 9d

Step 2: Find the terms:

It is given that, Sum of the 4th and 8th terms of AP is 24:

Which implies, a4 + 98 = 24

Putting values we get:

  • a + 3d + a + 7d = 24
  • 2a + 10d = 24

Now we will divide by 2 into both sides, and we get:

a + 5d = 12 ….. (1)

It is also given that the sum of the 6th and 10th terms of AP is 44:

a6 + a10 = 44

Putting values we get:

a + 5d + a + 9d = 44

Simplifying further:

2a + 14d = 44

Dividing by 2 into both sides, we get:

a + 7d = 22 ….. (2)

Step 3: Determine the common difference

Solving equations (1) and (2)

we get, -2d = -10

which further simplifies to:

d = -10/-2 = 5

Therefore, d = 5 or common difference = 5

Step 4: Determine the terms of integers

From (1) we have, a + 5d = 12

Putting the values of d we get:

  • a = 12 - 5d
  • a = 12 - 25
  • a = -13

The first term of AP, a = -13

Or, a1 = -13

The second term of AP:

a2 = a + (2 - 1)d

= a + d

= - 13 + 5

= -8

The third term of AP:

a3 = a + (3 - 1)d

= a + 2d

= - 13 + 2 x 5

= - 13 + 10

= - 3

a1 = -13, a2 = -8, a3 = -3

Therefore, the three integers are -13, -8, and -3.

Summary:

The sum of the 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 44. Find the first three terms of the A.P.

The sum of the 4th and 8th terms of an A.P. is 24 and the sum of the 6th and 10th terms is 44. The first three terms of the A.P. are -13, -8, and -3.

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