If the terms of a sequence with even integers are a1, a2, a3, . . . ,

1 gives us nothing about the type of sequence, so insufficient
2 tells us its AM, insufficient
By combining, in order to find sum of AM, we need a1,n,d. By 2 we know d=2 but n and a1 are unknown, hence E

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This is a moderately difficult question on sequences. There are, maybe a few traps laid for the student who is not careful, so it’s better to tread a safe path by not making any assumptions.

To start off with, from the question statement, we know that all the numbers of the sequence \(a_1\), \(a_2\), \(a_3\),……, \(a_n\) are even integers. At this stage, for some of you, it might be very easy to fall into the trap of assuming that they are consecutive even integers, but, please resist the temptation; they need not be.
We are supposed to find the value of n i.e. the number of terms in the sequence.

Using statement I alone, we know the sum of the terms in the sequence is 2178. But, we cannot say anything about the number of terms in the sequence. For example, there can be 3 terms in the sequence i.e. 724, 726 and 728. Observe that 2178 is a multiple of 3, so I tried to divide it into 3 equal parts and then manipulate the numbers on the extremeties.
The sequence can also have 2 terms – 2 and 2176.
Since we do not have any other information about the terms, statement I alone is insufficient. Options A and D can be ruled out. Possible answers at this stage are B, C and E.

Using statement II alone, we know that the terms in the sequence are consecutive integers. However, we do not know their sum. We already saw that, when the sum is given, it’s hard to calculate the number of terms. With the sum not being given, it’s impossible to find out a unique value for the number of terms. So, statement II alone is insufficient. Option B can be eliminated.

Combining statements I and II, we know that the sum of consecutive even integers is 2178. But, this can be obtained in many ways. For example, there can be 3 terms – 724, 726 and 728. Or, they can be 9 terms – 234, 236, 238,240, 242, 244, 246, 248, 250.

Both statements are insufficient when taken together. Option C can be eliminated. So, the correct answer option is E.

Hope this helps!

1. 2178 = (n/2) {2a + (n-1)d}; we don't know a and d. Insufficient.

2. d=2, Insufficient.

1+2 will give:

2178 = (n/2) {2a + (n-1)2}; still a is missing. E

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