The sum of the first 10th term of an a.p is 255,the sum of the first 20 term is 1010.find the next 2020 term of the progression
I want further explanation
10/2[2a+(10-1)d=255 20/2[2a+(20-1)d=1010. So a=3, and d=5. Then the 2020 term is 3+(2020-1)5.And the answer is 10098.
Yes
What of the next 20 terms
Model question paper
To find the next term of an arithmetic progression (a.p.), we need to identify the common difference (d) between consecutive terms. Once we have the common difference, we can use the formula for the nth term of an a.p. to find the next term.
In this case, we are given information about the sum of the first 10 terms and the sum of the first 20 terms:
Sum of the first 10 terms = 255
Sum of the first 20 terms = 1010
To find the common difference, we can subtract the sum of the first 10 terms from the sum of the first 20 terms:
Sum of the next 10 terms = Sum of the first 20 terms - Sum of the first 10 terms
Sum of the next 10 terms = 1010 - 255
Sum of the next 10 terms = 755
Now we have the sum of the next 10 terms (755), and we know that the number of terms (n) in an a.p. is given by n = 10.
The formula for the sum of the first n terms of an a.p. is:
Sn = (n/2)(2a + (n-1)d)
where Sn is the sum of the first n terms, n is the number of terms, a is the first term, and d is the common difference.
By rearranging the formula, we can solve for the common difference (d):
d = (2(Sn) - n(2a)) / (n^2 - n)
Substituting the given values:
d = (2(755) - 10(2a)) / (10^2 - 10)
d = (1510 - 20a) / 90
Now, we can use the formula for the nth term of an a.p. to find the next term:
an = a + (n-1)d
Substituting the given values and solving for the next term:
a2020 = a + (2020 - 1)d
To find a, we can use the formula for the sum of the first n terms:
Sn = (n/2)(2a + (n-1)d)
255 = (10/2)(2a + (10-1)d)
255 = 5(2a + 9d)
51 = 2a + 9d
Similarly, for the sum of the first 20 terms:
1010 = (20/2)(2a + (20-1)d)
1010 = 10(2a + 19d)
101 = 2a + 19d
Now we have a system of two equations with two variables (a and d):
51 = 2a + 9d ...(1)
101 = 2a + 19d ...(2)
We can solve this system of equations to find the values of a and d.
Subtracting equation (1) from equation (2):
(101 - 51) = (2a - 2a) + (19d - 9d)
50 = 10d
d = 5
Substituting the value of d back into equation (1):
51 = 2a + 9(5)
51 = 2a + 45
2a = 51 - 45
2a = 6
a = 3
Now that we have the values of a and d, we can find the next term:
a2020 = a + (2020 - 1)d
a2020 = 3 + (2019)(5)
a2020 = 3 + 10095
a2020 = 10098
Therefore, the next term of the arithmetic progression is 10098.