Number Sequence Generator

def arithmetic_nth(a, d, n):
    """n-th term of arithmetic sequence (1-indexed)."""
    return a + (n - 1) * d

def arithmetic_sum(a, d, n):
    """Sum of first n terms of arithmetic sequence."""
    return n * (2 * a + (n - 1) * d) / 2

def geometric_nth(a, r, n):
    """n-th term of geometric sequence (1-indexed)."""
    return a * (r ** (n - 1))

def geometric_sum(a, r, n):
    """Sum of first n terms of geometric sequence."""
    if r == 1:
        return a * n
    return a * (1 - r ** n) / (1 - r)

def fibonacci(n):
    """Generate first n Fibonacci numbers."""
    seq = [0, 1]
    for _ in range(n - 2):
        seq.append(seq[-1] + seq[-2])
    return seq[:n]

def detect_sequence(seq):
    """Auto-detect if sequence is arithmetic, geometric, or other."""
    diffs = [seq[i+1] - seq[i] for i in range(len(seq) - 1)]
    if len(set(diffs)) == 1:
        return "arithmetic", diffs[0]
    ratios = [seq[i+1] / seq[i] for i in range(len(seq) - 1) if seq[i] != 0]
    if len(set(round(r, 9) for r in ratios)) == 1:
        return "geometric", ratios[0]
    return "other", None

# Examples
print(arithmetic_nth(2, 3, 5))       # 14  (2,5,8,11,14)
print(arithmetic_sum(1, 1, 100))     # 5050 (Gauss sum)
print(geometric_sum(1, 0.5, float("inf")))  # use formula: 1/(1-0.5)=2
print(geometric_sum(1, 2, 10))       # 1023
print(fibonacci(10))                 # [0,1,1,2,3,5,8,13,21,34]
print(detect_sequence([3,6,12,24]))  # ('geometric', 2.0)