Progressions

General Term of Arithmetic Progression

1. If pth term, qth term, rth term of an arithmetic progression are in arithmetic progression, show that p, q, r are in arithmetic progression.

2. If mth and nth terms of an arithmetic progression be equal to n and m respectively, find its pth term, where p equals mn.

General Term of Geometric Progression

3. Natural numbers p, q, r, s are in arithmetic progression. If P, Q, R, S be respectively the pth, qth, rth and sth terms of a geometric progression, show that P, Q, R, S are in geometric progression.

4. Let a, b, c, d be distinct real numbers and S(x) be a quadratic expression in real variable x, with the sum of the squares of a, b, c as leading coefficient, the sum of the squares of b, c, d as constant term and -2(ab+bc+cd) as the coefficient of x. If S(x) is not positive for any value of x, show that a, b, c, d are in geometric progression.

5. Let f(x)=2x+1. Show that the unequal numbers f(x), f(2x), f(4x) can be in geometric progression for no value of x.

General Term of Harmonic Progression

6. Let T(r) denotes the rth term of a progression. If T(r)-T(r+1) bears a constant ratio with T(r).T(r+1), show that T(1), T(2), T(3), ... are in harmonic progression.

7. Let T(r) denotes the rth term of a progression. If the ratio of T(2).T(3) to T(1).T(4) and that of T(2)+T(3) to T(1)+T(4), both be equal to three times the ratio of T(2)-T(3) to T(1)-T(4); show that T(1), T(2), T(3), ... are in harmonic progression.

8. Show that the square roots of any three prime numbers cannot be any three terms of an arithmetic, a geometric or a harmonic progression.

General Term of Arithmetic Progression

1. If pth term, qth term, rth term of an arithmetic progression are in arithmetic progression, show that p, q, r are in arithmetic progression.

2. If mth and nth terms of an arithmetic progression be equal to n and m respectively, find its pth term, where p equals mn.

General Term of Geometric Progression

3. Natural numbers p, q, r, s are in arithmetic progression. If P, Q, R, S be respectively the pth, qth, rth and sth terms of a geometric progression, show that P, Q, R, S are in geometric progression.

4. Let a, b, c, d be distinct real numbers and S(x) be a quadratic expression in real variable x, with the sum of the squares of a, b, c as leading coefficient, the sum of the squares of b, c, d as constant term and -2(ab+bc+cd) as the coefficient of x. If S(x) is not positive for any value of x, show that a, b, c, d are in geometric progression.

5. Let f(x)=2x+1. Show that the unequal numbers f(x), f(2x), f(4x) can be in geometric progression for no value of x.

General Term of Harmonic Progression

6. Let T(r) denotes the rth term of a progression. If T(r)-T(r+1) bears a constant ratio with T(r).T(r+1), show that T(1), T(2), T(3), ... are in harmonic progression.

7. Let T(r) denotes the rth term of a progression. If the ratio of T(2).T(3) to T(1).T(4) and that of T(2)+T(3) to T(1)+T(4), both be equal to three times the ratio of T(2)-T(3) to T(1)-T(4); show that T(1), T(2), T(3), ... are in harmonic progression.

8. Show that the square roots of any three prime numbers cannot be any three terms of an arithmetic, a geometric or a harmonic progression.