the fifth term of an arithmetic progression is 19,and the fourteenth term is 55. Find its first term
"the fifth term of an arithmetic progression is 19"
----> a+4d = 19
" the fourteenth term is 55"
----> a+13d=55
subtract the two equations:
9d = 36
d = 4
in a+13d = 55
a + 13(4) = 55
a = 3
To find the first term of the arithmetic progression, we need to determine the common difference.
The formula to find the nth term of an arithmetic progression is:
An = A1 + (n - 1)d
where:
An = nth term
A1 = first term
n = term number
d = common difference
We are given that the fifth term (A5) is 19 and the fourteenth term (A14) is 55.
Using the formula, we can write the equations:
A5 = A1 + (5-1)d
19 = A1 + 4d ... (1)
A14 = A1 + (14-1)d
55 = A1 + 13d ... (2)
We can solve this system of equations to find the values of A1 and d.
Subtracting equation (1) from equation (2), we get:
55 - 19 = (A1 + 13d) - (A1 + 4d)
36 = 9d
d = 4
Now, substitute the value of d back into equation (1) to find A1:
19 = A1 + 4(4)
19 = A1 + 16
A1 = 19 - 16
A1 = 3
Therefore, the first term of the arithmetic progression is 3.
Correct
Why did the arithmetic progression go to therapy? Because it couldn't understand why it was feeling, well, so "ordinary"!
But let's solve this riddle anyways! We can use the formula for the nth term of an arithmetic progression, given by:
a_n = a + (n-1)d,
where a is the first term, n is the term number, and d is the common difference.
We're given that the fifth term is 19, so plugging into the formula:
a_5 = a + (5-1)d,
19 = a + 4d.
We're also given that the fourteenth term is 55:
a_14 = a + (14-1)d,
55 = a + 13d.
Now, we have a system of two equations:
19 = a + 4d,
55 = a + 13d.
To solve this system, we can subtract the first equation from the second equation to eliminate a:
55 - 19 = a + 13d - (a + 4d),
36 = 13d - 4d,
36 = 9d.
Dividing both sides by 9:
4 = d.
Now, substitute this value of d into the first equation:
19 = a + 4(4),
19 = a + 16,
a = 19 - 16,
a = 3.
So, the first term of the arithmetic progression is 3. Congratulations, you now know the "first" of the matter!
To find the first term of the arithmetic progression, we can use the formula:
An = A1 + (n - 1)d
where An is the nth term, A1 is the first term, and d is the common difference.
Given that the fifth term (A5) is 19 and the fourteenth term (A14) is 55, we can use these values to form the following equations:
A5 = A1 + (5 - 1)d
19 = A1 + 4d ...............(1)
A14 = A1 + (14 - 1)d
55 = A1 + 13d ...............(2)
We now have a system of two equations with two unknowns (A1 and d). We can solve this system of equations to find the values of A1 and d.
From equation (1), we get:
A1 = 19 - 4d ...............(3)
Substitute equation (3) into equation (2):
55 = (19 - 4d) + 13d
55 = 19 + 9d
55 - 19 = 9d
36 = 9d
d = 36/9
d = 4
Now, substitute the value of d into equation (3):
A1 = 19 - 4(4)
A1 = 19 - 16
A1 = 3
Therefore, the first term (A1) of the arithmetic progression is 3.