4,30,120,340,780

Question

the next number in the pattern

Answer 1

Assume f(n)=a₀+a₁n+a₂n²+a₃n³+a₄n⁴ where 0≤n≤4.

f(0)=a₀=4;

f(1)=a₀+a₁+a₂+a₃+a₄=30, a₁+a₂+a₃+a₄=26;

f(2)=a₀+2a₁+4a₂+8a₃+16a₄=120, 2a₁+4a₂+8a₃+16a₄=116;

f(3)=a₀+3a₁+9a₂+27a₃+81a₄=340, 3a₁+9a₂+27a₃+81a₄=336;

f(4)=a₀+4a₁+16a₂+64a₃+256a₄=780, 4a₁+16a₂+64a₃+256a₄=776;

(1) f(2)-2f(1)=2a₂+6a₃+14a₄=64, a₂+3a₃+7a₄=32;

(2) f(3)-3f(1)=6a₂+24a₃+78a₄=258, a₂+4a₃+13a₄=43;

(3) f(4)-4f(1)=12a₂+60a₃+252a₄=672, a₂+5a₃+21a₄=56;

(4) (2)-(1)=a₃+6a₄=11;

(5) (3)-(2)=a₃+8a₄=13;

(5)-(4)=2a₄=2, a₄=1; a₃=5, a₂=32-15-7=10, a₁=26-10-5-1=10

f(n)=4+10n+10n²+5n³+n⁴ for n≥0.

f(n)=4+10(n-1)+10(n-1)²+5(n-1)³+(n-1)⁴ for n≥1.

Using this formula we get the series: 4, 30, 120, 340, 780, 1554, 2800, 4680, ...