The 10th common term between the series
3+7+11+… and 1+6+11+… is
(A) 191
(B) 193
(C) 211
(D) None of these
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Step-by-Step Solution
Step 1: Identify the given arithmetic progressions
We are given two arithmetic progressions (APs). For the first AP, the first term a1 is 3 and the common difference d1 is 7−3=4. For the second AP, the first term a2 is 1 and the common difference d2 is 6−1=5.
Step 2: Find the first common term
We list out the terms of both series to find the first common term. The terms of the first series are 3, 7, 11, 15, 19, 23, ... The terms of the second series are 1, 6, 11, 16, 21, 26, ... The first term that appears in both series is 11.
Step 3: Determine the common difference of the common terms
The common terms of two arithmetic progressions also form an arithmetic progression. The common difference of this new AP is the Least Common Multiple (LCM) of the common differences of the original two APs. Here, d1=4 and d2=5, so the LCM of 4 and 5 is 20.
Step 4: Formulate the AP of common terms
Now we have the first term of the AP of common terms, which is 11, and its common difference, which is 20. So, the AP of common terms is 11, 31, 51, ...
Step 5: Calculate the 10th common term
To find the 10th common term, we use the formula for the n -th term of an AP: Tn=a+(n−1)D. Substituting a=11, n=10, and D=20, we get T10=11+(10−1)20=11+9×20=11+180=191.