An arithmetic sequence has u₆ = -8 and u₁₃ = -22. Calculate the first term, u₁.
Answer (2)
The first term of the arithmetic sequence is 2. This was calculated using the formulas for the terms of the sequence and the given values for u₆ and u₁₃. We found the common difference and substituted it back to determine u₁.
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To solve for the first term u 1 of the arithmetic sequence, we need to use the general formula for any term in an arithmetic sequence:
u n = u 1 + ( n − 1 ) d
where u n is the n -th term, u 1 is the first term, d is the common difference, and n is the term number.
We are given:
u 6 = − 8 and u 13 = − 22
Let's use these to find the common difference d first. We can set up two equations using the formula:
For u 6 : u 6 = u 1 + ( 6 − 1 ) d = − 8 u 1 + 5 d = − 8
For u 13 : u 13 = u 1 + ( 13 − 1 ) d = − 22 u 1 + 12 d = − 22
Now, we solve these two equations simultaneously to find d and u 1 .
Subtract the first equation from the second:
( u 1 + 12 d ) − ( u 1 + 5 d ) = − 22 + 8 7 d = − 14 d = − 2
Now that we have the value of d , substitute it back into one of the original equations to find u 1 :
u 1 + 5 ( − 2 ) = − 8 u 1 − 10 = − 8 u 1 = 2
Therefore, the first term u 1 of the arithmetic sequence is 2.
