the 40th term of an arithmetic sequence is equal to the sum of the 20th and 31st term. if the common difference of the sequence is-10. find the first term.
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"the 40th term of an arithmetic sequence is equal to the sum of the 20th and 31st term"
----> a + 39d = a+19d + a+30d
a = -10d
but d = -10
a = 100
the first term is 100
To solve this problem, we'll start by understanding how an arithmetic sequence works. An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant.
Let's denote the first term of the arithmetic sequence as 'a' and the common difference as 'd'. In this case, the common difference is given as -10.
Now, we're given that the 40th term of the sequence is equal to the sum of the 20th and 31st terms. We can express this as an equation:
a + (40 - 1) * d = (a + (20 - 1) * d) + (a + (31 - 1) * d)
Simplifying the equation, we get:
a + 39d = a + 19d + a + 30d
Combining like terms, we have:
a + 39d = 2a + 49d
Subtracting a + 49d from both sides, we get:
39d = a
So, the first term 'a' is equal to 39 times the common difference 'd'. We know that in this case, d = -10. So we can substitute this value in the equation to find the first term:
a = 39 * (-10)
a = -390
Therefore, the first term of the arithmetic sequence is -390.