MIXED QUESTIONS ON ARITHMETIC AND GEOMETRIC PROGRESSION

(1)  Write the first 6 terms of the sequences whose nth terms are given below and classify them as arithmetic progression, geometric progression, arithmetico-geometric progression, harmonic progression and none of them.

(i)  1/2ⁿ⁺¹

(ii)  (n + 1)(n + 2) / (n + 3)(n + 4)

(iii) 4 (1/2)ⁿ

(iv) (−1)ⁿ/n

(v) (2n+3) / (3n+4)

(vi) 2018

(vii) (3n−2)/(3ⁿ⁻¹)    Solution

(2)  Write the first 6 terms of the sequences whose n^th term an is given below.

(i) a_n = n + 1 if n is odd

            n if n is even               Solution

(ii) a_n = 1 if n = 1

        2 if n = 2

a_n−1 + a_n−2 if n > 2             Solution

(iii)  a_n =  n if n is 1, 2 or 3

         a_n−1 + a_n−2 + a_n−3 if n > 3                Solution

(3)  Write the nth term of the following sequences.

(i)  2, 2, 4, 4, 6, 6, ................         Solution

(ii)  1/2 , 2/3 , 3/4 , 4/5 , 5/6, ...        Solution

(iii)  1/2 , 3/4 , 5/6 , 7/8 , 9/10, ...        Solution

(iv)  6, 10, 4, 12, 2, 14, 0, 16, −2, . .        Solution

(4) The product of three increasing numbers in GP is 5832. If we add 6 to the second number and 9 to the third number, then resulting numbers form an AP. Find the numbers in GP.          Solution

(5)  Write the nth term of the sequence

3/1²2², 5/2²3², 7/3²4² , . . . as a difference of two terms.      Solution

(6)  If t_k is the kth term of a GP, then show that t_n−k, t_n, t_n+kalso form a GP for any positive integer k.    Solution

(7)  If a, b, c are in geometric progression, and if a^1/x = b^1/y = c^1/z, then prove that x, y, z are in arithmetic progression.                Solution

(8)  The AM of two numbers exceeds their GM by 10 and HM by 16. Find the numbers.         Solution

(9)  If the roots of the equation (q − r)x² + (r − p)x + p − q = 0 are equal, then show that p, q and r are in AP.   Solution

(10)  If a, b, c are respectively the pth, qth and rth terms of a GP, show that (q − r) log a + (r − p) log b + (p − q) log c = 0.   Solution

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