Let a1, a2, a3, ldots be a G P of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a1 a9+a2 a4 α9+a5+a7 is equal to

Question

Let a1, a2, a3, ldots be a G P of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a1 a9+a2 a4 α9+a5+a7 is equal to  (A)  (B)  (C)  (D)

Tardigrade Admin · Accepted Answer

a 4 ⋅ a 6=9 ⇒( a 5)2=9 ⇒ a 5=3 a 5+ a 7=24 ⇒ a 5+ a 5 r 2=24 ⇒(1+ r 2)=8 ⇒ r =√7 ⇒ a =(3/49) ⇒ a 1 a 9+ a 2 a 4 a 9+ a 5+ a 7=9+27+3+21=60

Q. Let $a_{1}, a_{2}, a_{3}, \dots$ be a $GP$ of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$ , then $a_{1} a_{9} + a_{2} a_{4} α_{9} + a_{5} + a_{7}$ is equal to ____

$a_{4} \cdot a_{6} = 9 \Rightarrow (a_{5})^{2} = 9 \Rightarrow a_{5} = 3$
$& a_{5} + a_{7} = 24 \Rightarrow a_{5} + a_{5} r^{2} = 24 \Rightarrow (1 + r^{2}) = 8 \Rightarrow r = 7$
$\Rightarrow a = \frac{3}{49}$
$\Rightarrow a_{1} a_{9} + a_{2} a_{4} a_{9} + a_{5} + a_{7} = 9 + 27 + 3 + 21 = 60$

Q. Let a1​,a2​,a3​,… be a GP of increasing positive numbers. If the product of fourth and sixth terms is 9 and the sum of fifth and seventh terms is 24, then a1​a9​+a2​a4​α9​+a5​+a7​ is equal to ____

a4​⋅a6​=9⇒(a5​)2=9⇒a5​=3 &a5​+a7​=24⇒a5​+a5​r2=24⇒(1+r2)=8⇒r=7​ ⇒a=493​ ⇒a1​a9​+a2​a4​a9​+a5​+a7​=9+27+3+21=60

Q. Let $a_{1}, a_{2}, a_{3}, \dots$ be a $GP$ of increasing positive numbers. If the product of fourth and sixth terms is $9$ and the sum of fifth and seventh terms is $24$ , then $a_{1} a_{9} + a_{2} a_{4} α_{9} + a_{5} + a_{7}$ is equal to ____

$a_{4} \cdot a_{6} = 9 \Rightarrow (a_{5})^{2} = 9 \Rightarrow a_{5} = 3$
$& a_{5} + a_{7} = 24 \Rightarrow a_{5} + a_{5} r^{2} = 24 \Rightarrow (1 + r^{2}) = 8 \Rightarrow r = 7$
$\Rightarrow a = \frac{3}{49}$
$\Rightarrow a_{1} a_{9} + a_{2} a_{4} a_{9} + a_{5} + a_{7} = 9 + 27 + 3 + 21 = 60$