Arithmetic progression in one shot for quick revision

If p^("th"), 2p^("th") and 4p^("th") terms of an arithmetic progression are in geometric progression, then the common ratio of the geometric progression is

If the 2^(nd),5^(th) and 9^(th) terms of a non-constant arithmetic progression are in geometric progession, then the common ratio of this geometric progression is

If the sum of the first 100 terms of an arithmetic progression is -1 and the sum of the even terms is 1, then the 100^("th") term of the arithmetic progression is

The first term of an arithmetic progression is 1 and the sum of the first nine terms equal to 369 . The first and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression.

The first term of an arithmetic progression is 1 and the sum of the first nine terms equal to 369 . The first and the ninth term of a geometric progression coincide with the first and the ninth term of the arithmetic progression. Find the seventh term of the geometric progression.

Let {a _(1),a _(2), a _(3)......} be a strictly increasing sequence of positive integers in an arithmetic progression. Prove that there is an infinite subsequence of the given sequence whose terms are in a geometric progression.

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. 12 ,\ 2,\ -8,\ -18 ,\ ... (ii) 3,\ 3,\ 3,\ 3,\ ddot (iii) p ,\ p+90 ,\ p+180 ,\ p+270 ,\ where p=(999)^(999)

Find out which of the following sequences are arithmetic progressions. For those which are arithmetic progressions, find out the common difference. 3,\ 6,\ 12 ,\ 24 ,\ (ii) 0,\ -4,\ -8,\ -12 ,\ ddot (iii) 1/2,1/4,1/6,1/8,\

If S_(n)=n^(2)a+(n)/(4)(n-1)d is the sum of the first n terms of an arithmetic progression, then the common difference is

If x, y, z are in arithmetic progression and a is the arithmetic mean of x and y and b is the arithmetic mean of y and z, then prove that y is the arithmetic mean of a and b.