(a) How many numbers between 75 and 500 are divisible by 7? (b) The 8th term of an Arithmetic Progression (A.P) is 5 times the 3rd term while the 7th term i...
Question 1 Report
(a) How many numbers between 75 and 500 are divisible by 7?
(b) The 8th term of an Arithmetic Progression (A.P) is 5 times the 3rd term while the 7th term is 9 greater than the 4th term. Write the first 5 terms of the A.P.
Answer Details
(a) The multiples of \(7\) between \(75\) and \(500\): the first is \(77 = 7\times 11\) and the last is \(497 = 7\times 71.\) \[\text{count} = 71 - 11 + 1 = \mathbf{61}.\]
(b) Let the first term be \(a\) and common difference \(d.\)
8th term is \(5\) times the 3rd: \(a + 7d = 5(a + 2d) \Rightarrow a + 7d = 5a + 10d \Rightarrow 4a + 3d = 0.\)
7th term is \(9\) greater than the 4th: \((a + 6d) - (a + 3d) = 9 \Rightarrow 3d = 9 \Rightarrow d = 3.\)
Then \(4a + 3(3) = 0 \Rightarrow 4a = -9 \Rightarrow a = -\tfrac{9}{4} = -2.25.\)
First five terms (adding \(d = 3\) each time): \[\mathbf{-2\tfrac{1}{4},\ \tfrac{3}{4},\ 3\tfrac{3}{4},\ 6\tfrac{3}{4},\ 9\tfrac{3}{4}}.\]
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