An arithmetic sequence has u₆ = -8 and u₁₃ = -22. Calculate the first term, u₁.

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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Answered by Anonymous | 2025-07-04

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.

Answered by MasonWilliamTurner | 2025-07-06