the first and the last terms of an A.P are 6 and 171 in that order if there are 14 terms find the 16th term of the A.P
To find the 16th term of an arithmetic progression (A.P.), we need to know the common difference (d).
Given:
First term (a₁) = 6
Last term (aₙ) = 171
Number of terms (n) = 14
We can use the formula to find the common difference (d):
aₙ = a₁ + (n - 1)d
Substituting the given values, we have:
171 = 6 + (14 - 1)d
Simplifying the equation:
171 = 6 + 13d
Subtracting 6 from both sides:
165 = 13d
Dividing both sides by 13:
d = 165 / 13
d = 12.69231 (rounded to 5 decimal places)
Now that we know the common difference (d), we can find the 16th term (a₁₆) using the formula:
a₁₆ = a₁ + (16 - 1)d
Substituting the values:
a₁₆ = 6 + (16 - 1) * 12.69231
Simplifying the equation:
a₁₆ = 6 + 15 * 12.69231
Calculating:
a₁₆ = 6 + 190.38466
a₁₆ ≈ 196.38466
Therefore, the 16th term of the arithmetic progression is approximately 196.38466.
To find the 16th term of an arithmetic progression (A.P.), we need to first determine the common difference.
In this case, we know that the first term is 6 (a₁) and the last term is 171 (aₙ). We are also given that there are 14 terms (n = 14).
We can use the formula for the nth term of an A.P. to find the common difference:
aₙ = a₁ + (n-1)d
Substituting the known values:
171 = 6 + (14-1)d
To solve for d, we simplify the equation:
171 = 6 + 13d
165 = 13d
d = 165/13
d = 12.692307692307692
Now that we have the common difference (d), we can find the 16th term (a₁₆) by using the same formula:
a₁₆ = a₁ + (16-1)d
Substituting the values:
a₁₆ = 6 + (16-1)(12.692307692307692)
a₁₆ = 6 + (15)(12.692307692307692)
a₁₆ = 6 + 190.38461538461538
a₁₆ ≈ 196.38
Therefore, the 16th term of the A.P. is approximately 196.38.
a1 = 6
a14 = a1 + 13 d = 171
6 + 13 d = 171
Subtract 6 to both sides
13 d = 165
d = 165 / 13
a16 = a14 + 2 d
a16 = 171 + 2 ∙ 165 / 13 = 171 + 330 / 13 =
2223 / 13 + 330 / 13 = 2553 / 13