Let 3, 6, 9, 12, … upto 78 terms and 5, 9, 13, 17, … upto 59 terms be two series. Then, the sum of terms common to both the series is equal to _____.
Correct Answer: 2223
Solution and Explanation
The correct answer is 2223
Given : 3,6,9,12 ..... upto 78 terms
5,9,13,17 ...... upto 59 terms
Now , last term of first series t78 \(= 3+77×3=234\)
and last term of second series t59 \(= 5+58×4=237\)
Now , common difference of common terms = LCM {3,4} = 12
Therefore , First common term is 9 and last common term is 225
So, series will be 9,21,33 ....... 225
\(∵ t_n = 225\)⇒ 9 + (n-1)12 = 225
⇒ n = 19
∴ Sum of common on terms = \(S_n = \frac{19}{2} (9+225) = 2223\)
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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