A.P Assignment - X

Formulae for Sum of A.P.

Find the sum of

(i) 3 +7 +11 +15 ... to 30 terms

(ii) 1 +2/3 +1/3 +0 +.. to 19 terms

(iii) 5/2, 10/3, 25/6, 5 ... upto 24 terms

(iv) 51 +50 +49 +... +21

(v) 72 +70 +68 +... +40.

Find the sum of an A.P. of

(i) 25 terms whose nth term is 2n+5

(ii) 19 terms whose nth term is (2n+1)/3

If the sum of n terms of a series is an²+bn, where a, b are constants, show that it is an A.P. Find the first term and the common difference. Also find an A.P. whose sum of any number of terms is equal to the square of the number of terms.

[Hint. an²+bn=n² for all n  => a = 1, b = 0.]

Of a, l, d, n, Sn determine the ones which are missing for the following arithmetic progressions:

(i) a = -2, d = 5, Sn = 568

(ii) l = 8, n = 8, S8 = -20

(iii) d = 2/3, l = 10, n = 20.

(i) How many terms of series 13 +11 +9 +... make the sum 45? Explain the double answer.

(ii) How many terms of the sequence 18, 16, 14,... should be taken so that their sum is zero?

(i) Determine the sum of first 35 terms of an A.P. if T2 = 2, T7 = 22.

(ii) It the third term of an A.P. is 1 and 6th term is -11, find the sum of first 32 terms.

(iii) If the first term of an A.P. is 2 and the sum of first 4 terms is equal to one fourth of the sum of the next five terms, find the sum of first 30 terms.

If the sum of an n terms of two arithmetic series are in ratio

(i) (14 -4n):(3n +5), find the ratio of their 8th terms

(ii) (2 +3n):(3 +2n), find the ratio of their 7th terms.

(i) If the ratio of sum of m terms of an A.P. to the sum of n terms is m² : n², show that ratio of the pth and qth term is (2p -1) : (2q -1).

(ii) The sums of m and n terms of an A.P. are in ratio (2m +1) : (3n +1). Find the ratio of its 7th and 10th terms.

(i) If in an A.P., S1 = 6 and S7 = 105, prove that

    Sn : Sn -3 : : (n +3) : (n -3).

(ii) If in an A.P., S3 = 6 and S6 = 3, prove that

      2(2n +1)Sn +4 = (n +4)S2n +1.

(i) Find the sum of all two digit numbers.

(ii) Find the sum of all natural numbers between 100 and 1000 which are divisible by 2 as well as by 5.

(iii) Find the sum of all two digit numbers which leave 1 as remainder when divided by 3.

(iv) Find the sum of all odd integers between 2 and 100 which are divisible by 3.

[Hint. (iv) The odd integers between 2 and 100 which are divisible by 3 are 3, 9, 15, 21, ..., 99.]

(i) The sum of three numbers in A.P. is -3 and their product is 8. Find the numbers.

(ii) The sum of three consecutive numbers in an A.P. is 24 and the sum of their squares is 194. Find the numbers.

(iii) The sum of three numbers in an A.P. is 30, and the ratio of first to third is 3 : 7. Find the numbers.

(iv) Find five numbers in an A.P. whose sum is 25 and the ratio of the first to the last is 2 : 3.

(v) Divide 32 into four parts which are in A.P. such that the product of extremes is to the product of means is 7 : 15.