determine the A.P whose fourth term is 18 and the difference of the ninth term from the fifteenth term is 30.
An arithmetic progression (A.P) or arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.
If the initial term of an arithmetic progression is a1 and the common difference of successive members is d, then the nth term of the sequence is given by:
a n = a 1 + ( n - 1 ) d
a 1 = first term
In this case:
a 8 = a 1 + ( 8 - 1 ) d
a 8 = a 1 + 7 d = 18
a 15 = a 1 + ( 15 - 1 ) d
a 15 = a 1 + 14 d
a 9 = a 1 + ( 9 - 1 ) d
a 9 = a 1 + 8 d
a 15 - a 9 = ( a 1 + 14 d ) - ( a 1 + 8 d )
a 15 - a 9 = a 1 + 14 d - a 1 - 8 d
a 15 = a 1 - a1 + 14 d - 8 d
a 15 - a 9 = 6 d
a 15 - a 9 = 30
6 d = 30 Divide both sides by 6
d = 30 / 6
d = 5
a 8 = a 1 + 7 d = 18
a 1 + 7 * 5 = 18
a 1 + 35 = 18
a 1 = 18 - 35
a 1 = - 17
A.P
a n = - 17 + ( n - 1 ) * 5
- 17 , - 12 , - 7 , - 2 , 3 , 8 , 13 , 18 , 23 , 28 , 33 , 38 , 43 , 48 , 53
Good enough to understand 💁👍👍
It is given that 4th term is 18
a4=18
a+3d =18
a=18-3d.......(1)
Also, a15 -a9 =a+14d-1-8d=30
6d= 30
d=5
From (1), we get
a= 18-15=3
Therefore the A.P is
3,8,13,18,23
It is given that 4th term is 18 a4=18 a+3d=18
a=18-3d -(1)
a15-a9=a+14d-1-8d=30
6d=30
d=5
a=18-15=3
A.P is
3,8,13,18,23.....
To determine the arithmetic progression (A.P), we need to find the common difference (d) and the first term (a₁).
Let's start by finding the common difference (d):
We are given that the fourth term (a₄) is 18. In an arithmetic progression, the nth term (aₙ) can be calculated as follows: aₙ = a₁ + (n - 1)d
By substituting the values, we have:
a₄ = a₁ + (4 - 1)d
18 = a₁ + 3d
Next, we are given that the difference between the ninth term (a₉) and the fifteenth term (a₁₅) is 30:
By applying the same formula, we have:
a₉ = a₁ + (9 - 1)d
a₁₅ = a₁ + (15 - 1)d
The difference is given as 30:
a₉ - a₁₅ = 30
(a₁ + 8d) - (a₁ + 14d) = 30
-6d = 30
d = -5 (-6d/6 = 30/6)
Now that we have found the common difference (d = -5), we can substitute it back into the equation we derived from the fourth term:
18 = a₁ + 3(-5)
18 = a₁ - 15
a₁ = 18 + 15
a₁ = 33
Therefore, the first term (a₁) is 33 and the common difference (d) is -5. Hence, the arithmetic progression (A.P) is: 33, 28, 23, 18, 13, 8, ...