The first term of A.P. of 20th terms is 2and its last term is 59 . Find its 6th term from the end .
Nice
To find the 6th term from the end of an arithmetic progression (A.P.), we need to determine the total number of terms in the A.P.
Given:
- The first term (a₁) is 2.
- The last term (aₙ) is 59.
Since we know the first and last terms, we can use the formula to find the nth term in an A.P.:
aₙ = a₁ + (n - 1)d,
where aₙ is the last term, a₁ is the first term, n is the total number of terms, and d is the common difference between the terms.
From the given information, we can determine the total number of terms (n):
aₙ = a₁ + (n - 1)d
59 = 2 + (n - 1)d
To simplify the equation, we subtract 2 from both sides:
57 = (n - 1)d
To further simplify, we can observe that 57 is the (n - 1)th multiple of the common difference (d).
Since the common difference is the same for an arithmetic progression, we can find the value of d by subtracting the first term (a₁) from the last term (aₙ):
d = (aₙ - a₁) / (n - 1)
d = (59 - 2) / (n - 1)
d = 57 / (n - 1)
Now, we have both the expressions for (n - 1)d and d:
(n - 1)d = 57
d = 57 / (n - 1)
We can set these two expressions equal to each other:
57 = 57 / (n - 1)
To solve for n, we isolate it by multiplying both sides by (n - 1):
57(n - 1) = 57
Simplifying the equation:
57n - 57 = 57
57n = 57 + 57
57n = 114
Dividing both sides by 57:
n = 114 / 57
n = 2
Therefore, the total number of terms (n) in the arithmetic progression is 2. Since we are asked to find the 6th term from the end, we subtract 6 from the total number of terms:
6th term from the end = 2 - 6
6th term from the end = -4
The 6th term from the end of the arithmetic progression is -4.
59 = 2+19d
Now you can find d.
6th from the end is T15 = 2+14d.