What is the sum of the first four terms of sequence S?

WHAT IS THE BEST APPROACH TO SOLVE SUCH PROBLEM ?

According to statement 1 = x + y + (x+y)/2 + [(x+y)/2 + y]/2 = S

No info about Sum, NS.

St 2- x+y+z = 30.

NS.

On combining,

x + y + (x+y)/2 = 30 => x+y = 20.

We can find the 3rd term with this info but since we have no idea what type of sequence it is w.r.t first three terms, we cannot determine x & y.

FYI, 3rd term will be (x+y)/2 = 10. (replacing x+y with 20). But, can you find x & y ? (x+y =20 ...x & y could be multiple values)

What is the sum of the first four terms of sequence S?

(1) After the first two terms of S,the value of each term of S is equal to the average (arithmetic mean) of the last two preceding terms.

(2) The average (arithmetic mean) of the first three terms of S is 10.


Let first 2 terms be a,b
As per statement 1, sequence is: a, b, (a+b)/2, b +(a+b)/2 = a, b, (a+b)/2, (a+3b)/2
Sum = (4a + 6b)/2 (Sum can be anything, since we don't know values of a and b, so eliminate option A and D)

Let first 3 terms be a,b,c
As per statement 2, a+b+c = 3*10 = 30
Now, 4th term can be any term, thus eliminate option B

Combining both statements:
a + b + (a+b)/2 = 30
a+b = 20
Sum = (4a+6b)/2 = [(4a+4b) + 2b] / 2
Sum = 2(a+b) + b
Sum = 2*20 + b = 40 + b
Since, we don't know the value of b, still not sufficient, Select Option E

Solution:

Question Stem Analysis:

We need to determine the sum of the first four terms of sequence S. If we can determine the values of the four terms, then we can determine their sum.

Statement One Alone:

Since we don’t know the value of any of the 4 terms, statement one is not sufficient.

Statement Two Alone:

Even though we don’t know the value of any of the first 3 terms, we know their sum is 30. However, since we don’t know the 4th term, statement two is not sufficient.

Statements One and Two Together:

We can let a and b be the first two terms, then using the first statement, the 3rd term is (a + b)/2 = a/2 + b/2, and the 4th term is (b + (a + b)/2)/2 = b/2 + a/4 + b/4 = a/4 + 3b/4. From the second statement, we can create the equation:

a + b + a/2 + b/2 = 30

3a/2 + 3b/2 = 30

3a + 3b = 60

a + b = 20

However, since we don’t know the individual values of a and b, we can’t determine the 4th term. For example, if a = 8 and b = 12, then the 4th term is a/4 + 3b/4 = 2 + 9 = 11 and the sum of the first 4 terms is 30 + 11 = 41 (recall that the sum of the first 3 terms is 30). However, if a = 12 and b = 8, then then the 4th term is a/4 + 3b/4 = 3 + 6 = 9 and the sum of the first 4 terms is 30 + 9 = 39. Both statements together are not sufficient.

Answer: E

Can anyone try this non algebraically? I'm a little confused why it isn't C.