Question

若等比数列an的各项均为正数且a4a9十a5a8十a6a7=300,则lga1+lga2十,,,

Answer 1

是求lga1＋lga2＋lga3＋…＋lga12吧？

易知：a1a12＝a1^2q^11
a2a11＝a1q·a1q^10＝a1^2q^11
a3a10＝a1q^2·a1q^9＝a1^2q^11
a4a9＝a1q^3·a1q^8＝a1^2q^11
a5a8＝a1q^4·a1q^7＝a1^2q^11
a6a7＝a1q^5·a1q^6＝a1^2q^11
∴a1a12＝a2a11＝a3a10＝a4a9＝a5a8＝a6a7
由已知：a4a9十a5a8十a6a7＝300
∴3·a1^2q^11＝300
∴a1^2q^11＝100
∴a1a12＝a2a11＝a3a10＝a4a9＝a5a8＝a6a7＝100
∴lga1＋lga2＋lga3＋…＋lga12
＝(lga1＋lga12)＋(lga2＋lga11)＋(lga3＋lga10)＋(lga4＋lga9)＋(lga5＋lga8)＋(lga6＋lga7)
＝lg(a1a12)＋lg(a2a11)＋lg(a3a10)＋lg(a4a9)＋lg(a5a8)＋lg(a6a7)
＝lg100＋lg100＋lg100＋lg100＋lg100＋lg100
＝12