The first term and the last term of an AP are 6 and 171 in that order. If there are 14th term find the 16th term of AP
We are given that the first term of the arithmetic progression (AP) is 6 and the last term is 171. We know that the nth term of an AP is given by the formula:
an = a + (n-1)d
Where:
an is the nth term
a is the first term
d is the common difference
We can rewrite the formula as:
an = a + (n-1)(a - an)/n
Using the given values and solving for the common difference, we have:
171 = 6 + (14-1)(6 - 171)/14
170 = 13(165)/14
170 = 2145/14
14 * 170 = 2145
2380 = 2145
Now we can find the 16th term of the AP using the formula.
a16 = 6 + (16-1)(6 - 171)/14
a16 = 6 + 15(6 - 171)/14
a16 = 6 + 15(-165)/14
a16 = 6 - 2475/14
a16 = 6 - 175
a16 = -169
Therefore, the 16th term of the AP is -169.
To find the 14th term of an arithmetic progression (AP), we need to use the formula for the nth term of an AP which is:
An = A + (n - 1)d
where An is the nth term, A is the first term, n is the term number, and d is the common difference.
Given that the first term (A) is 6 and the 14th term (An) is to be found, we can use the formula to find the common difference (d):
An = A + (n - 1)d
14th term (An) = 6 + (14 - 1)d
Now we substitute the values:
An = 6 + 13d
We also know that the last term of the AP is 171. Using the same formula, substitute An = 171:
171 = 6 + 13d
Simplify the equation:
165 = 13d
Divide both sides by 13:
d = 165/13
d = 12.69
Now that we have found the common difference (d), we can find the 16th term (An) using the same formula:
An = A + (n - 1)d
16th term (An) = 6 + (16 - 1)(12.69)
16th term (An) = 6 + 15(12.69)
16th term (An) = 6 + 190.35
16th term (An) = 196.35
Therefore, the 16th term of the arithmetic progression is 196.35.
To find the 14th term of an arithmetic progression (AP) given the first term and the last term in that order, we need to find the common difference (d) between the terms. Once we have the common difference, we can find any term in the AP using the formula:
Term(n) = First Term + (n - 1) * Common Difference
Given:
First term (a) = 6
Last term (l) = 171
We know that the nth term of an AP is given by:
Term(n) = a + (n - 1) * d
We can substitute the values of the first term (a = 6) and last term (l = 171) into the formula to find the common difference (d):
6 + (n - 1) * d = 171
To solve for d, subtract 6 from both sides:
(n - 1) * d = 165
Now, we need to find the value of n. Since the 14th term is required, we have:
(n - 1) = 14
n = 14 + 1
n = 15
Substituting the value of n = 15 into the equation:
14 * d = 165
To find the common difference (d), divide both sides by 14:
d = 165 / 14
d ≈ 11.7857
Now, we have the common difference (d) ≈ 11.7857. We can find the 16th term (Term(16)) by substituting the values into the formula:
Term(16) = 6 + (16 - 1) * 11.7857
Simplifying,
Term(16) = 6 + 15 * 11.7857
Term(16) ≈ 182.7857
Therefore, the 16th term of the arithmetic progression (AP) is approximately 182.7857.