Members

A geometric sequence with six members has the sum of all six members equal to 63; the sum of the even members (that has an even index) has a value of 42. Find these members.

Correct answer:

a₁ =  1
a₂ =  2
a₃ =  4
a₄ =  8
a₅ =  16
a₆ =  32

Step-by-step explanation:

$$\begin{gathered} a_1+q{a}_1+q^2{a}_1+q^3{a}_1+q^4{a}_1+q^5{a}_1=63 \\ q{a}_1+q^3{a}_1+q^5{a}_1=42 \\ {a}_1+q^2{a}_1+q^4{a}_1=63-42=21 \\ \frac{a_1+q{a}_1+q^2{a}_1+q^3{a}_1+q^4{a}_1+q^5{a}_1}{q{a}_1+q^3{a}_1+q^5{a}_1}=\frac{63}{42} \\ \frac{1+q+q^2+q^3+q^4+q^5}{q+q^3+q^5}=\frac{63}{42} \\ 2(1+q+q^2+q^3+q^4+q^5)=3(q+q^3+q^5) \\ 2+2q^2+2q^4=q+q^3+q^5 \\ q=2 \\ {a}_1=21/(1+q^2+q^4)=21/(1+2^2+2^4)=1 \end{gathered}$$

$a_2=q\cdot a_1=2\cdot 1=2$

$a_3=q\cdot a_2=2\cdot 2=4$

$a_4=q\cdot a_3=2\cdot 4=8$

$a_5=q\cdot a_4=2\cdot 8=16$

$$\begin{gathered} a_6=q\cdot a_5=2\cdot 16=32 \\ \text{ Verifying Solution: } \\ {s}_1=a_1+a_2+a_3+a_4+a_5+a_6=1+42+4+8+16+32=63 \\ {a}_2=a_2+a_4+a_6=42+8+32=42 \end{gathered}$$

Try another example