The sum of the common terms of the following three arithmetic progressions\(3,7,11,15, \ldots , 399\), \(2,5,8,11, \ldots , 359\)and \(2,7,12,17, \ldots , 197,\) is equal to _____
The sum of the common terms of the following three arithmetic progressions\(3,7,11,15, \ldots , 399\), \(2,5,8,11, \ldots , 359\)and \(2,7,12,17, \ldots , 197,\) is equal to _____
Show Hint
When solving problems involving the intersection of arithmetic progressions, find the LCM of the common differences to identify possible common terms efficiently.
Correct Answer: 321
Solution and Explanation
The given arithmetic progressions (APs) are:
\[\text{AP}_1 : 3, 7, 11, 15, \ldots, 399 \quad (d_1 = 4),\]
\[\text{AP}_2 : 2, 5, 8, 11, \ldots, 359 \quad (d_2 = 3),\]
\[\text{AP}_3 : 2, 7, 12, 17, \ldots, 197 \quad (d_3 = 5).\]
The least common multiple (LCM) of the common differences \(d_1, d_2, d_3\) is:
\[\text{LCM}(4, 3, 5) = 60.\]
The first common term of the three sequences can be found by checking the terms that satisfy all three APs. The common terms are:
\[47, 107, 167.\]
The sum of the common terms is:
\[47 + 107 + 167 = 321.\]
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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