The numbers 1, 4, 16 can be three terms (not necessarily consecutive) of no A.P. only on G.P. infinite number o A.P.’s infinite number of G.P.’s

If p,q and r ( pneq ) are terms ( not necessarily consecutive) of an A.P., then prove that there exists a rational number k such that (r-q)/(q-p) =k. hence, prove that the numbers sqrt2,sqrt3 and sqrt5 cannot be the terms of a single A.P. with non-zero common difference.

For what value of p are 2p+1,\ 13 ,\ 5p-3 are three consecutive terms of an A.P.?

The numbers t (t ^(2) +1) , -1/2t ^(2) and 6 are three consecutive terms of an A.P. If t be real, then find the the next two terms of A.P.

If 4/5,\ a ,\ 2 are three consecutive terms of an A.P., then find the value of a .

In an infinite G.P. the sum of first three terms is 70. If the externme terms are multipled by 4 and the middle term is multiplied by 5, the resulting terms form an A.P. then the sum to infinite terms of G.P. is :

If second, third and sixth terms of an A.P. are consecutive terms of a G.P. write the common ratio of the G.P.

Find three numbers a,b,c between 2 & 18 such that; their sum is 25; the numbers 2,a,b are consecutive terms of an AP & the numbers b,c,18 are consecutive terms of a G.P.

If the sum of three consecutive numbers in A.P., is 24 and their product is 440, find the numbers.

If second third and sixth terms of an A.P. are consecutive terms o a G.P. write the common ratio of the G.P.

Three numbers are in A.P. and their sum is 15. If 1,4 and 19 be added to these numbers respectively the number are in G.P. Find the numbers .