Find the first and second difference of x^4 -6x^3+11x^2-5x+8 with n=1 show that the fourth difference is constant.
Answers
Answer:
elow is an AP. Find the first term, common difference and next term of each.
(i) 9, 15, 21, 27, ...
(ii) 11, 6, 1, −4, ...
(iii) −1,
-5
6
,
-2
3
,
-1
2
, ...
(iv)
√
2
,
√
8
,
√
18
,
√
32
, ...
(v)
√
20
,
√
45
,
√
80
,
√
125
, ...
ANSWER:
(i) The given progression 9, 15, 21, 27, ... .
Clearly, 15 − 9 = 21 − 15 = 27 − 21 = 6 (Constant)
Thus, each term differs from its preceding term by 6. So, the given progression is an AP.
First term = 9
Common difference = 6
Next term of the AP = 27 + 6 = 33
(ii) The given progression 11, 6, 1, −4, ... .
Clearly, 6 − 11 = 1 − 6 = −4 − 1 = −5 (Constant)
Thus, each term differs from its preceding term by −5. So, the given progression is an AP.
First term = 11
Common difference = −5
Next term of the AP = −4 + (−5) = −9
(iii) The given progression −1,
-5
6
,
-2
3
,
-1
2
, ...
Clearly,
-5
6
-(-1)=
-2
3
-(
-5
6
)=
-1
2
-(
-2
3
)=
1
6
(Constant)
Thus, each term differs from its preceding term by
1
6
. So, the given progression is an AP.
First term = −1
Common difference =
1
6
Next term of the AP =
-1
2
+
1
6
=
-2
6
=
-1
3
(iv) The given progression
√
2
,
√
8
,
√
18
,
√
32
, ...
This sequence can be re-written as
√
2
, 2
√
2
, 3
√
2
, 4
√
2
, ...
Clearly, 2
√
2
-
√
2
=3
√
2
-2
√
2
=4
√
2
-3
√
2
=
√
2
(Constant)
Thus, each term differs from its preceding term by
√
2
. So, the given progression is an AP.
First term =
√
2
Common difference =
√
2
Next term of the AP = 4
√
2
+
√
2
=5
√
2
=
√
50
(v) The given progression
√
20
,
√
45
,
√
80
,
√
125
, ...
This sequence can be re-written as 2
√
5
, 3
√
5
, 4
√
5
, 5
√
5
, ...
Clearly, 3
√
5
-2
√
5
=4
√
5
-3
√
5
=5
√
5
-4
√
5
=
√
5
(Constant)
Thus, each term differs from its preceding term by
√
5
. So, the given progression is an AP.
First term = 2
√
5
=
√
20
Common difference =
√
5
Next term of the AP = 5
√
5
+
√
5
=6
√
5
=
√
180
Step-by-step explanation:
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given a polynomial , find its first and second difference and show that fourth difference is constant
Explanation:
- let the polynomial be given as,
- now differences are formed by the values of 'y' from corresponding values of 'x' given that 'x' values are in arithmetic progression.
- hence, below is values for 'y' for given 'x' values in AP,
---->ANSWER
---->ANSWER
----->constant differences