The first term of an Arithmetic progression is -5, find it's common difference if its last term is 31
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so the last term is 36 more than the first. Assuming the difference is an integer, that means it is one of the factors of 36.
To find the common difference of an arithmetic progression, we need to use the formula:
nth term = first term + (n-1) * common difference
Here, the first term is given as -5 and the last term as 31. Let's use this information to solve for the common difference.
We can substitute the values into the formula:
31 = -5 + (n-1) * common difference
Simplifying the equation, we get:
31 + 5 = (n-1) * common difference
36 = (n-1) * common difference
Now, we need to find the value of n. We know that the last term is given as 31, so we can use the formula for the nth term to solve for n:
nth term = first term + (n-1) * common difference
Substituting the values, we get:
31 = -5 + (n-1) * common difference
Rearranging the equation, we have:
(n-1) * common difference = 31 - (-5)
(n-1) * common difference = 36
Since common difference cannot be zero, we know that (n-1) must be a factor of 36. Let's list the factors of 36: 1, 2, 3, 4, 6, 9, 12, 18, and 36.
Now, we need to check which of these factor combinations would give us an integer value for n. We know that the number of terms in an arithmetic progression is given by n. So, if we find a pair of factors that subtract to give an odd number (since we have an odd number of terms in this case), then we have found our solution.
Let's go through the factors:
For 1: (n-1) = 1, (common difference) = 36. But this doesn't give us an odd number.
For 2: (n-1) = 2, (common difference) = 18. This gives us an odd number, as (n-1) - 2 = 1.
Therefore, the common difference of the arithmetic progression is 18.