If n arithmetic means are inserted between a and 100 such that the ratio of the first mean to the last mean is 1 : 7 and a + n = 33, then the value of n is:
21
22
23
24
The Correct Option is C
Solution and Explanation
The correct answer is (C) : 23
a, A1, A2 …….. An, 100
Let d be the common difference of above A.P. then
\(\frac{a+d}{100−d}=\frac{1}{7}\)
⇒ 7a + 8d = 100 …(i)
and a + n = 33 …(ii)
and 100 = a + (n + 1)d
⇒\(100=a+(34−a)\frac{(100−7a)}{8}\)
⇒ 800 = 8a + 7a2 – 338a + 3400
⇒ 7a2 – 330a + 2600 = 0
\(⇒a=10,\frac{260}{7}\)
but
\( a≠\frac{260}{7}\)
∴ n = 23
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Concepts Used:
Arithmetic Progression
Arithmetic Progression (AP) is a mathematical series in which the difference between any two subsequent numbers is a fixed value.
For example, the natural number sequence 1, 2, 3, 4, 5, 6,... is an AP because the difference between two consecutive terms (say 1 and 2) is equal to one (2 -1). Even when dealing with odd and even numbers, the common difference between two consecutive words will be equal to 2.
In simpler words, an arithmetic progression is a collection of integers where each term is resulted by adding a fixed number to the preceding term apart from the first term.
For eg:- 4,6,8,10,12,14,16
We can notice Arithmetic Progression in our day-to-day lives too, for eg:- the number of days in a week, stacking chairs, etc.
Read More: Sum of First N Terms of an AP