Q1 If the pth term of an AP is 1/q and th qth term 1/p Prove that - Maths - Arithmetic Progressions

Question

Q1 If the pth term of an AP is 1/q and th qth term 1/p Prove
that the sum of the first pq terms is 1/2 ( pq + 1)
Q2 Find the common difference of an AP whose first term is 100
and the sum of whose first 6 terms is 5 times the sum of the next 6
terms
Q3 Find the sum of the first 20 terms of an AP, in which the 3rd
term is 7 and 7th term is 2 more than thrice of its 3rd term
Q4 how many terms of the series 51, 47, 43, 39, be taken to make
the sum of 330 ?
Plz answer as soon as possible

Lalit Mehra · Accepted Answer

Dear Student! 2 Let d be the common difference of A P First term of A P , a = 100 Let Sn denote the sum of first n terms of the A P Given $(a_{1} + a_{2} + a_{3} + a_{4} + a_{5} + a_{6}) = 5 (a_{7} + a_{8} + a_{9} + a_{10} + a_{11} + a_{12})$ Adding 5 (a1 + a2 + a3 + a4 + a5 + a6) to both sides, we get $6 (a_{1} + a_{2} + a_{3} + + a_{6}) = 5 (a_{1} + a_{2} + + a_{12}) ∴ 6 S_{6} = 5 S_{12} \Rightarrow 6 \times \frac{6}{2} [2 \times 100 + (6 - 1) \times d] = 5 \times \frac{12}{2} [2 \times 100 + (12 - 1) \times d] \Rightarrow 18 (200 + 5 d) = 30 (200 + 11 d) (∵ S_{n} = \frac{n}{2} [2 a + (n - 1) d]) \Rightarrow 3 (200 + 5 d) = 5 (200 + 11 d) \Rightarrow 600 + 15 d = 1000 + 55 d \Rightarrow 55 d - 15 d = 600 - 1000 \Rightarrow 40 d = - 400 \Rightarrow d = - 10$ Thus, the common difference of A P is â€“10 We are happy to see the interest you are showing towards this section! Thank you for your response I have already posted the answer to your four questions However, due to a paucity of time we will not be able to immediately answer all your queries Try to solve the remaining questions If you have any problem in solving the questions, then get back to us we will be happy to help you Cheers!