Question 1184673
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(i) 5th term of the ap: a+4d
9th term: a+8d
16th term: a+15d<br>
In a gp, the square of any term is equal to the product of the terms before and after it:<br>
{{{(a+8d)^2=(a+4d)(a+15d)}}}
{{{a^2+16ad+64d^2=a^2+19ad+60d^2}}}
{{{4d^2-3ad=0}}}
{{{d(4d-3a)=0}}}<br>
The problem is of little interest if d=0, so<br>
{{{4d-3a=0}}}
{{{4d=3a}}}
{{{d=(3/4)a}}}<br>
(i) ANSWER: d = (3/4)a<br>
(ii) 21st term: a+20d = a+15a = 16a
37th term: a+36d = a+27a = 28a
65th term: a+64d = a+48a = 49a<br>
28a/16a = 7/4; 49a/28a = 7/4; this sequence is geometric with common ratio 7/4<br>