All the terms of the sequence have to be non-negative integers As soon as we get a negative term it would mean that the sequence terminates at the previous term. Let's write the first few terms:
t3=150−t2
t4=2t2−150
t5=300−3t2
t6=5t2−450
t7=750−8t2
t8=13t2−1200
t9=1950−21t2
t10=34t2−3150
Now let's try to make as many of them positive as possible: 150−t2≥0or150≥t2 2t2−150≥0ort2≥75 300−3t2≥0or100≥t2 5t2−450≥0 or t2≥90 750−8t2≥0 or 93.75≥t2 13t2−1200≥0 or t2≥92.30 So t2 must be greater than 92 and less than 94, for the first 8 terms to be positive So when t2 = 93. the sequence would have exactly 8 terms. For every other value of t2, the number of terms would be less than 8. So the answer is 93.