Question

已知等比数列an满足：a1+a2+a3+a4+a5=3,则a1-a2+a3-a4+a5的值是

Answer 1

设公比为q,q≠1

∵a1+a2+a3+a4+a5=3

a1²+a2²+a3²+a4²+a5²=12

（a1²,a2²,a3²,a4²,a5² 公比为q²)

∴a1(1-q^5)/(1-q)=3 (1)

a1²(1-q^10)/(1-q^2)=12

==> a1²[(1+q^5)(1-q^5)]/[(1+q)(1-q)] =12 【平方差公式 1-q^10=(1+q^5)(1-q^5)】

==> a1(1+q^5)/(1+q) *a1(1-q^5)/(1-q)=12

==> a1(1+q^5)/(1+q) × 3=12

∴a1(1+q^5)/(1+q) =4

∵a1,-a2,a3,-a4,a5为公比为-q的等比数列

∴a1-a2+a3-a4+a5

=a1[1-(-q)^5]/(1+q)

=a1(1+q^5)/(1+q) =4