Decoding the Series Pattern

The problem states that the series is generated by representing the decimal number 99 in different, successive number bases. Let's verify this pattern for the given terms.

• Term 1 (99): This is the representation of 99 in base 10. ($9 \times 10^1 + 9 \times 10^0 = 99$)
• Term 2 (90): This is the representation of 99 in base 11. ($9 \times 11^1 + 0 \times 11^0 = 99$)
• Term 3 (83): This is the representation of 99 in base 12. ($8 \times 12^1 + 3 \times 12^0 = 99$)
• Term 4 (78): This is the representation of 99 in base 13. ($7 \times 13^1 + 8 \times 13^0 = 99$)

The pattern is confirmed. To find the next terms, we must continue this sequence of converting the decimal number 99 into base 14 and then base 15.

Calculating the Next Terms

Next Term in the Series (Base 14)

To convert the decimal value 99 to base 14, we perform division with remainder:

$99 \div 14 = 7$ with a remainder of $1$.

So, the base-14 representation of 99 is written as 71. The next term is 71.

Second Next Term in the Series (Base 15)

To convert the decimal value 99 to base 15, we again perform division with remainder:

$99 \div 15 = 6$ with a remainder of $9$.

So, the base-15 representation of 99 is written as 69. The second next term is 69.

Conclusion

The next two terms in the series are 71, 69.