Question

For the sequence 1, −2, 3, −4, 5, −6, 7, . . ., what’s the difference between the mean of the sequence’s
first 400 terms and the mean of its first 200 terms?

Answer 1

Separate that into 2 different sequences.

1, -2, 3, -4, 5, -6, 7 ...


=> 1, 3, 5, 7 ... [a = 1, d = 2]

=> -2, -4, -6, -7 ... [a = -2, d = -2]

Now you have 2 different arithmetic sequences and hence, can find the answer! :D

Answer 2

What do I do with these two sequences?

Answer 3

@AkashdeepDeb

Answer 4

I am not sure if by mean they mean arithmetic mean here.

Answer 5

Do you know the arithmetic mean of an arithmetic progression?

Answer 6

I think they mean just the mean of the first 400 terms of that sequence

Answer 7

Well, the first 400 terms of the sequence contains, 200 of the first sub-sequence and 200 of the second sub-sequence right?

Answer 8

correct

Answer 9

Do you know how to find the sum of an arithmetic sequence?

Answer 10

No

Answer 11

But, I looked up a formula

Answer 12

Well, you do need to know that I suppose. I am not aware of any other way to find the sum of all terms then.

Okay, now that you know the formula, which is:
\[S = \frac{n}{2} [2a + (n-1)d]\]
Where,
S = sum
n = no. of terms
a = first term of any sequence
d = common difference

NOTE:

If you are not aware of arithmetic progressions (or sequences) please don't solve it this way.