The 7th term of an a.p is 11 and the nth term is 3. Find the first five term
a+6d=11
a+(n-1)d=3
subtract them:
6d - (n-1)d = 8
6d - nd + d = 8
7d - nd = 8
d = 8/(7-n) , it should be noted that n ≠ 7
back in the 1st:
a + 48/(7-n) = 11
a = 11 - 48/(7-n)
= (77 - 11n - 48)/(7-n)
= (29 - 11n)/(7-n)
so the terms are:
(29 - 11n)/(7-n) , (37 - 11n)/(7-n), ... etc
check:
term(7) = a + 6d
= (29 - 11n)/(7-n) + 48/(7-n)
= (56 - 8n)/(7-n)
= (77 - 11n) /(7-n)
= 11(7-n)/(7-n) = 11 , yeahhhh
To find the first five terms of an arithmetic progression (a.p), we need to determine the common difference (d) first.
Given that the 7th term (a7) is 11 and the nth term (an) is 3, we can utilize these equations:
a7 = a1 + 6d (Equation 1)
an = a1 + (n-1)d (Equation 2)
From Equation 1, if the 7th term (a7) is 11, it can be written as:
11 = a1 + 6d
Similarly, from Equation 2, if the nth term (an) is 3, it can be written as:
3 = a1 + (n-1)d
Now we have two equations with two variables (a1 and d), we can solve them simultaneously.
Let's solve for a1 by writing Equation 2 in terms of a1:
a1 = an - (n-1)d
Substituting the values of an = 3 and n = 1 into the equation:
a1 = 3 - (1-1)d
a1 = 3
Now we can substitute the value of a1 = 3 into Equation 1:
11 = 3 + 6d
Rearranging the equation:
6d = 11 - 3
6d = 8
d = 8/6
d = 4/3
Therefore, the common difference (d) is 4/3.
Now we have a1 = 3 and d = 4/3. We can find the first five terms of the arithmetic progression (a.p) using the formula:
an = a1 + (n-1)d
Substituting the values, we get:
a2 = a1 + (2-1)d = 3 + (2-1)(4/3) = 3 + 4/3 = 13/3
a3 = a1 + (3-1)d = 3 + (3-1)(4/3) = 3 + 8/3 = 19/3
a4 = a1 + (4-1)d = 3 + (4-1)(4/3) = 3 + 12/3 = 15/1 = 15
a5 = a1 + (5-1)d = 3 + (5-1)(4/3) = 3 + 16/3 = 25/3
Therefore, the first five terms of the arithmetic progression are:
a1 = 3
a2 = 13/3
a3 = 19/3
a4 = 15
a5 = 25/3
To find the first five terms of the arithmetic progression (AP), we need to determine the common difference first.
Let's start by finding the common difference, denoted by "d."
Given:
The 7th term of the AP is 11, so we can say that a(7) = 11.
The nth term of the AP is 3, so we can say that a(n) = 3.
We can use the formula for the nth term of an AP to express these terms in terms of the common difference:
a(n) = a(1) + (n-1) * d ...(1)
From equation (1), substitute the values of the 7th term and the nth term:
11 = a(1) + (7-1) * d --> 11 = a(1) + 6d ...(2)
3 = a(1) + (n-1) * d --> 3 = a(1) + (n-1) * d ...(3)
Now, we have two equations with two unknowns: a(1) and d.
To find the common difference (d), we can subtract equation (3) from equation (2):
11 - 3 = (a(1) + 6d) - (a(1) + (n-1) * d)
8 = 6d - (n-1) * d
8 = 6d - nd + d
8 = (6 - n)d + d
8 = (7 - n) * d
From this equation, we can find the value of d:
d = 8 / (7 - n) ...(4)
Now, substitute the value of d from equation (4) into equation (3) to find the value of a(1):
3 = a(1) + (n-1) * (8 / (7 - n))
3(7 - n) = a(1)(7 - n) + (n-1)(8)
21 - 3n = 7a(1) - a(1)n + 8 - 8n
21 - 3n + 8n + a(1)(n - 7) = 0
(a(1) - 29)n + 29 + 21 = 0
(a(1) - 29)n + 50 = 0
For the equation to hold true, the coefficient of n must be zero. Therefore,
a(1) - 29 = 0
a(1) = 29
Now that we have the value of a(1) and the common difference (d), we can find the first five terms of the AP.
The first term (a(1)) is 29, and the common difference (d) is given by equation (4), where we substitute in the value of n:
d = 8 / (7 - n)
Let's calculate the first five terms using the formula for the nth term of an AP:
Term 1: a(1) = 29
Term 2: a(2) = a(1) + d = 29 + d
Term 3: a(3) = a(1) + 2d = 29 + 2d
Term 4: a(4) = a(1) + 3d = 29 + 3d
Term 5: a(5) = a(1) + 4d = 29 + 4d
Now, substitute the value of d into each term:
Term 1: a(1) = 29
Term 2: a(2) = 29 + (8 / (7 - 2)) = 29 + (8 / 5) = 29 + 1.6 = 30.6
Term 3: a(3) = 29 + 2 * (8 / (7 - 3)) = 29 + (16 / 4) = 29 + 4 = 33
Term 4: a(4) = 29 + 3 * (8 / (7 - 4)) = 29 + (24 / 3) = 29 + 8 = 37
Term 5: a(5) = 29 + 4 * (8 / (7 - 5)) = 29 + (32 / 2) = 29 + 16 = 45
Therefore, the first five terms of the arithmetic progression are 29, 30.6, 33, 37, 45.