SOLUTION: The coefficients of the powers of x in the 2nd, 3rd and 4th term of the expansion {{{ (1 + x)^n }}} is in arithmetic progression where n is positive integer. Find the number of coe

Question 1184224: The coefficients of the powers of x in the 2nd, 3rd and 4th term of the expansion is in arithmetic progression where n is positive integer. Find the number of coefficients of the odd powers of x in the expansion?
Found 2 solutions by ikleyn, greenestamps:
Answer by ikleyn(52824)   (Show Source): You can put this solution on YOUR website!
.

            A quick solution is as follows.

Look at the Pascal's triangle in this Wikipedia article https://en.wikipedia.org/wiki/Pascal%27s_triangle You will see momentarily that the sought line is the 8th line for . Those coefficients are 7, 21, 35. The sum of coefficients at odd degrees is HALF of the total sum, which is = 128. THEREFORE, the answer to the problem's question is 128/2 = 64.

Solved.

Answer by greenestamps(13203)   (Show Source): You can put this solution on YOUR website!

The coefficients that are in arithmetic progression can be found algebraically; however, it is unclear what the question is asking for....

The coefficients in the expansion of (1+x)^n are...

1st term:
2nd term:
3rd term:
4th term:

The 2nd, 3rd, and 4th terms are in arithmetic progression, so the 3rd term is twice the sum of the 2nd and 4th terms:

n=7 or n=2

n=2 makes no sense in the problem because (1+x)^2 only has 3 terms.

So n=7.

The 2nd, 3rd, and 4th terms in the expansion of (1+x)^7 are 7, 21, and 35 -- in arithmetic progression with common difference 14.

But I don't know how to answer the question because I don't know what it means:

"Find the number of coefficients of the odd powers of x in the expansion."

The expansion of (1+x)^7 has 8 terms; half of them are even powers of x and half are odd powers.

So I guess the answer is 4.

But after doing some good math to find a value of n for which the coefficients of the 2nd, 3rd, and 4th terms in the expansion of (1+x)^n form an arithmetic progression, that seems like a very odd question to ask....

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