General term and middle term - One Line Questions
1.
For the expansion of (x - y)^n, the (r+1)th term has a sign determined by: —
(-1)^r
2.
In the expansion of (x - y)^n, the general term T_{r+1} is: —
(-1)^r C(n, r) x^{n-r} y^r
3.
For the expansion of (a + b)^n, if n is odd, the second middle term is the ((n+3)/2)th term. What is the power of 'a' in this term? —
(n-1)/2
4.
If n is even, the middle term in the expansion of (a + b)^n is C(n, n/2) a^{n/2} b^{n/2}. What is the term number? —
(n/2) + 1
5.
In the expansion of (a + b)^n, if n is odd, the two middle terms are symmetric. The powers of 'a' in these terms are: —
(n+1)/2 and (n-1)/2
6.
If n is odd, the two middle terms in the expansion of (a + b)^n have powers of 'b' as: —
(n+1)/2 and (n+3)/2
7.
What is the term number for the middle term when n=12 in the expansion of (a+b)^n? —
7th term
8.
Which term in the expansion of (x + y)^12 is the middle term? —
7th term
9.
What is the index of the term that contains a^3 b^7 in the expansion of (a + b)^10? —
8th term
10.
The general term T_{r+1} in the expansion of (a + b)^n corresponds to the term with: —
a^{n-r} b^r
11.
Find the middle term in the expansion of (x + 1)^10. —
C(10, 5) x^5
12.
The middle term in the expansion of (x + 1/y)^10 is: —
C(10, 5) x^5 (1/y)^5
13.
What is the coefficient of the middle term in the expansion of (x + y)^6? —
C(6, 3)
14.
What is the coefficient of the middle term in the expansion of (x + 3y)^6? —
C(6, 3) * 3^3
15.
In the expansion of (x + 1/x)^6, find the middle term. —
C(6, 3) x^0
16.
What is the coefficient of the term x^3 y^4 in the expansion of (x + y)^7? —
C(7, 4)
17.
Find the coefficient of x^5 in the expansion of (x + 2)^7. This involves finding a specific general term. —
C(7, 5) * 2^2
18.
What is the (r+1)th term in the expansion of (2x - 3y)^7? —
C(7, r) (2x)^{7-r} (-3y)^r
19.
In the expansion of (x + 1/x)^8, the middle term is: —
C(8, 4) x^0
20.
In the expansion of (a + b)^n, the middle term occurs when r = n/2 (if n is even). What is the coefficient? —
C(n, n/2)
21.
For the expansion of (x + y)^n, the coefficient of the (r+1)th term is always: —
C(n, r)
22.
For the expansion of (ax + by)^n, the (r+1)th term is: —
C(n, r) (ax)^{n-r} (by)^r
23.
What is the index 'r' in the general term T_{r+1} = C(n, r) x^{n-r} y^r of the expansion (x + y)^n? —
It ranges from 0 to n
24.
The general term in the expansion of (x + y)^n is T_{r+1}. For the term involving y^k, what is the value of r? —
k
25.
What is the index of the term containing a^{n-k}b^k in the expansion of (a+b)^n? —
k+1
26.
The general term in the expansion of (x + 1/x)^n is T_{r+1} = C(n, r) x^{n-r} (1/x)^r. For this term to be independent of x, the power of x must be 0. Therefore: —
n - 2r = 0
27.
In the expansion of (a + b)^n, the general term T_{r+1} has the power of 'b' as r. What is the power of 'a'? —
n-r
28.
The general term in the expansion of (ax + by)^n is C(n, r) (ax)^{n-r} (by)^r. The power of 'a' in this term is: —
n-r
29.
What is the value of 'r' for the middle term in the expansion of (a + b)^n when n is even? —
n/2
30.
In the expansion of (a + b)^n, if n is even, the middle term is the ((n/2) + 1)th term. The power of 'b' in this term is: —
n/2
31.
In the expansion of (a + b)^n, if n is even, how many middle terms are there? —
One
32.
The general term in the expansion of (x^p + y^q)^n is T_{r+1} = C(n, r) (x^p)^{n-r} (y^q)^r. What is the power of x in this term? —
p(n-r)
33.
The general term of the expansion (ax^p + by^q)^n is T_{r+1}. The power of x in this term is: —
p(n-r)
34.
If the expansion is (x - 2)^5, the general term is: —
T_{r+1} = C(5, r) x^{5-r} (-2)^r
35.
In the binomial expansion of (a + b)^n, what is the general term? —
T_{r+1} = C(n, r) a^{n-r} b^r
36.
The general term in the expansion of (x + 1/x)^n is: —
T_{r+1} = C(n, r) x^{n-r} (1/x)^r
37.
What is the middle term in the expansion of (2x - 3)^7? —
T_4 = -C(7, 3) (2x)^4 (3)^3
38.
For the expansion of (a - b)^7, the middle terms are: —
T_4 = -C(7, 3) a^4 b^3 and T_5 = C(7, 4) a^3 b^4
39.
What is the middle term in the expansion of (x - 2y)^6? —
T_4 = C(6, 3) x^3 (-2y)^3
40.
The middle term of (x + 2y)^8 is: —
T_5 = C(8, 4) x^4 (2y)^4
41.
In the expansion of (2x - 1)^9, the middle terms are: —
T_5 = C(9, 4) (2x)^5 (-1)^4 and T_6 = C(9, 5) (2x)^4 (-1)^5
42.
Find the middle term in the expansion of (3x - 4y)^10. —
T_6 = C(10, 5) (3x)^5 (-4y)^5
43.
The general term in the expansion of (a + b)^n is T_{r+1}. The term number is related to the power of 'b' by: —
Term Number = Power of b + 1
44.
If n is an even positive integer, the middle term in the expansion of (a + b)^n is: —
The ((n/2) + 1)th term
45.
If n is an odd positive integer, the middle terms in the expansion of (a + b)^n are: —
The ((n+1)/2)th term and the ((n+3)/2)th term
46.
The general term in the expansion of (x + y)^n is given by: —
The (r+1)th term
47.
For the expansion of (x + y)^10, the middle term is: —
The 6th term
48.
For the expansion of (x + y)^9, the middle terms are: —
The 5th term and the 6th term
49.
In the expansion of (a + b)^n, the term independent of x in (x + 1/x)^n is obtained when the powers of x cancel out. This happens when: —
The power of x in the first term equals the power of x in the second term
50.
If n is an odd positive integer, how many middle terms are there in the expansion of (a + b)^n? —
Two