The length of twelve pole from an AP if the third pole is 3m and the eight pole is 5m find the length of the first pole and the sum of the length of the pole
AAAaannndd the bot gets it wrong yet again!
a+2d = 3
a+7d = 5
so
S12 = 12/2 (2*11/5+11*2/5) = 52 4/5
To find the length of the first pole and the sum of the lengths of all twelve poles, we'll first need to determine the common difference (d) and then use it to calculate the first term (a) in the arithmetic progression (AP).
An arithmetic progression is a sequence of numbers in which the difference between consecutive terms is constant. In this case, the difference between terms will be the common difference (d).
We can identify the common difference (d) by subtracting the length of the third pole from the length of the eighth pole:
d = Length of the eighth pole - Length of the third pole
= 5m - 3m
= 2m
Now that we have the common difference, we can calculate the length of the first pole (a) by subtracting the product of the common difference and the number of terms before the first pole:
a = Length of the third pole - (2 * (3 - 1))
= 3m - (2 * 2)
= 3m - 4m
= -1m
Since the length of a pole cannot be negative, it's possible that there was a mistake in the given information or the sequence is not an arithmetic progression.
However, if we assume that there was an error and the second pole's length is actually 3m, we can recalculate the length of the first pole:
a = Length of the second pole - (2 * (2 - 1))
= 3m - (2 * 1)
= 3m - 2m
= 1m
Therefore, the length of the first pole is 1m.
To find the sum of the lengths of all twelve poles, we can use the formula for the sum of an arithmetic progression:
Sum = (n/2) * (2a + (n - 1)d),
where n is the number of terms, 2a is the sum of the first and last terms, and (n - 1)d is the sum of the remaining terms.
In this case, n = 12, a = 1m, and d = 2m:
Sum = (12/2) * (2 * 1m + (12 - 1) * 2m)
= 6 * (2m + 11 * 2m)
= 6 * (2m + 22m)
= 6 * 24m
= 144m
Therefore, the sum of the lengths of all twelve poles is 144m.
First pole = 1m
Sum of the length of the poles = 36m