Question

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

Answer 1

Hint: We will add the 4th and 8th terms of AP to get equation 1. Then, we add the 6th and 10th terms of AP to get equation 2. We solve those two equations to get the first term & common difference of an AP. From there, we will find the first, second & third term of the AP.

Complete step by step solution:

Formula: nth term of the AP (arithmetic progression) is given by the formula:-
an = a + (n − 1) d
where; a = first term of AP
and, d = common difference of 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) = a + 9d
It is given that, Sum of 4th & 8th term 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 on both sides, we get:
a + 5d = 12…………....eq -1
It is also given that the sum of 6th and 10th term of AP is 44
$\Rightarrow$ a6 + a10 = 44
Putting values we get,
a + 5d + a + 9d = 44
Simplifying further,
$\Rightarrow$2a + 14d = 44
Dividing by 2 on both sides, we get
$\Rightarrow$ a + 7d = 22…………….eq-2
Solving equations 1 and 2 we get,
−2d = −10 which further simplifies to,
$\Rightarrow$ $\text{d}=\dfrac{-10}{-2}=5$
Therefore, d = 5 or common difference = 5
∴ From eq-1 ,we have: a + 5d = 12
Putting the values of d we get,
$\Rightarrow$ a = 12 − 5d = 12 − 25 = −13
∴ a = −13
∴ first term of AP, a = −13
Or, a1 = −13
Second term of AP; a2 = a + (2 − 1)d
= a + d
= −13 + 5
= −8
Third term of AP; a3 = a + (3 − 1)d
= a + 2d
= −13 + 2 × 5
= −13 + 10
= −3
∴ a1 = −13, a2 = −8, a3 = −3
∴ first three term of AP are −13, −8, −3
Note: Here, we got the first term of AP as :
a1 = a = −13
We can also get second term of AP by adding common difference to the first term i.e.
∴ a2 = first term + common difference
= −13 + 5
= −8.
We can also get third term of AP by adding coming difference to second term i.e.
a3 = a2 + d = second term + common difference
= −8 + 5 = −3.