Asked by Lori
Find the 2nd and 3rd term of the sequence -7,__,__,-22,-23,...
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GPT-5 mini
AI
Let the first differences be d1, d2, d3, d4 (between terms 1–2, 2–3, 3–4, 4–5). We know t1 = −7, t4 = −22, t5 = −23, so t5 − t1 = −16 = d1 + d2 + d3 + d4.
If the sequence has constant second difference k, then d1, d2, d3, d4 form an arithmetic progression: d4 = d1 + 3k. Also d4 = −1 (since −23 − (−22) = −1). So
d1 + (d1+ k) + (d1+2k) + (d1+3k) = 4d1 + 6k = −16,
and d1 + 3k = −1.
Solving gives k = 2 and d1 = −7. Hence d2 = −5 and d3 = −3. Therefore
t2 = t1 + d1 = −7 + (−7) = −14,
t3 = t2 + d2 = −14 + (−5) = −19.
So the sequence is −7, −14, −19, −22, −23, ...
If the sequence has constant second difference k, then d1, d2, d3, d4 form an arithmetic progression: d4 = d1 + 3k. Also d4 = −1 (since −23 − (−22) = −1). So
d1 + (d1+ k) + (d1+2k) + (d1+3k) = 4d1 + 6k = −16,
and d1 + 3k = −1.
Solving gives k = 2 and d1 = −7. Hence d2 = −5 and d3 = −3. Therefore
t2 = t1 + d1 = −7 + (−7) = −14,
t3 = t2 + d2 = −14 + (−5) = −19.
So the sequence is −7, −14, −19, −22, −23, ...
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