If pth,qth,rth and sth terms of an A.P. are in G.P.,then show that (p-q),(q-r),(r-s) are also in G.P.(example of ncert).
plzzz explain it couldn't understand !!
ap, aq,ar, as are in G.P.
Now, using A.P. , we have ap = a + (p-1)d,
aq = a + (q-1)d
ar = a + (r-1)d
as = a + (s-1)d
Taking, a + (q – 1)d = k[a + (p – 1)d]
On subtracting a + (p – 1)d on both sides, we get
⇒ [a + (q – 1)d] – [a + (p – 1)d] = k[a + (p – 1)d] – [a + (p – 1)d],
Similarly, on taking a + (r – 1)d = k [a + (q – 1)d] and a + (s – 1)d = k [a + (r – 1)d] and subtracting a + (q – 1)d and a + (r – 1)d on both sides respectively, we get
[a + (r – 1)d] – [a + (q – 1)d] = k[a + (q – 1)d] – [a + (q – 1)d] and
–d(r – s) = (k – 1) [a + (r – 1)d]
On solving the respective equations, we get