Question

The 7th term, the 11t term and the 23rd term of an arithrietic sequence are the first three terms of a geometric sequence.(a) Find the common ratio of the geometric sequence.

(b) Find the 5th term of the arithmetic sequence.

Answer 1

Hi,

r=3

a_5=0

Explanation:

Let's assume for the arithmetic sequence:

• 
  `a_i` the i th term
• 
  `a` the first term
• d the common difference

`a_7=a+6*d\\a_(11)=a+10*d\\a_(23)=a+22*d\\`

for the geometric sequence:

• 
  `g_i` the i th term
• r the common ratio

`r=(u_(11))/(u_7) =(a+10d)/(a+6d) \\\\r^2=(u_(23))/(u_7) =(a+22d)/(a+6d) \\\\\\{((a+10d)/(a+6d) )}^2=(a+22d)/(a+6d) \\`

Cross-multiply:

`{({a+10d})}^2 =(a+22d)*(a+6d) \\\\`

Expand and simplify:

`a^2+20ad+100d^2=a^2+28ad+132d^2\\32d^2+8ad=0\\\\a=-4d`

(a) Find the common ratio of the geometric sequence.

`r=(a_11)/(a_7) =(a+10d)/(a+6d) =(6d)/(2d) =3\\\\`

b) Find the 5th term of the arithmetic sequence.

`a_5=a+4d=-4d+4d=0\\`