The sum of 16th term of an Ap is 240 and the next 4 terms is 220. Find the first term, common difference.
If your question means:
The sum of 16th term of an Ap is 240 and the sum of next 4 terms is 220.
then
Sum of first n terms of an AP:
Sn = n [ 2 a + ( n - 1 ) d ] / 2
In this case n = 16.
S16 = 240
16 ( 2 a + 15 d ) / 2 = 240
Multiply both sides by 2
16 ( 2 a + 15 d ) = 480
Divide both sides by 16
2 a + 15 d = 30
In AP
an = a + ( n - 1 ) d
a17 = a + 16 d
a18 = a + 17 d
a19 = a + 18 d
a20 = a + 19 d
The sum of next 4 terms:
a17 + a18 + a19 + a20 = 220
a + 16 d + a + 17 d + a + 18 d + a + 19 d = 220
4 a + 70 d = 220
Now you must solve system:
2 a + 15 d = 30
4 a + 70 d = 220
The solution is:
a = - 15 , d = 4
Of course:
a = first term
d = common difference.
sum of 16 terms is 240
8(2a + 15d) = 240
2a + 15d = 30
sum of 20 terms is 460 , (the first 240 plus the next 4 of 220 )
10(2a + 19d) = 460
2a + 19d = 46
subtract them: 4d = 16
d = 4
sub into 2a+15d = 30
2a + 60 = 30
a = -15
Yes
To find the first term (a) and the common difference (d) of an arithmetic progression (AP), we can use the formula for the nth term of an AP:
an = a + (n-1)d
where:
an = nth term
a = first term
n = position of the term
d = common difference
Given that the sum of the 16th term is 240, we can use the formula for the sum of the first n terms of an AP:
Sn = (n/2)(2a + (n-1)d)
where:
Sn = sum of the first n terms
We know that the sum of the 16th term is 240, so we can set up the equation:
240 = (16/2)(2a + (16-1)d)
240 = 8(2a + 15d)
30 = 2a + 15d
We are also given that the sum of the next four terms is 220. Using the same formula, we can set up another equation:
220 = (20/2)(2a + (20-1)d)
220 = 10(2a + 19d)
22 = 2a + 19d
Now we have a system of two equations:
30 = 2a + 15d
22 = 2a + 19d
Solving these equations simultaneously will give us the values of a and d.
To solve the system of equations, we can use the method of substitution or elimination. Let's use the elimination method:
Subtracting the second equation from the first, we get:
(30 - 22) = (2a + 15d) - (2a + 19d)
8 = -4d
d = -2
Substituting this value of d into either equation (let's use the first equation), we can solve for a:
30 = 2a + 15(-2)
30 = 2a - 30
60 = 2a
a = 30
Therefore, the first term (a) of the AP is 30 and the common difference (d) is -2.