Answer:
→ = = 1.4142...
→ = or = 1.5537....
→ = or = 1.5980....
As it is observed that = ; therefore, it can be understood that = as the trend has been observed as this until now.
Hence,
= = 1.6118...
= = 1.6161...
= = 1.6174...
= = 1.6178...
= = 1.6179...
= = 1.6179...
= = 1.6179...
[NOTE: VALUES OBTAINED USING THE GRAPHICAL DISPLAY CALCULATOR: CASIO fx – 9860G]
From this graph I understand that after the value keeps on increasing so does the value become similar i.e. 5 onwards the value is almost the same (1.62). the graph as seen faces a substantial rise in the start and the becomes straighter and consistent.
Q: find out the exact value for this surd ?
Answer:
=
As n becomes very large, - = 0 as n is
= = x.
x =
On squaring both sides –
=
→ = 1 + x
→ – x – 1 = 0
→
→
→
→
x= OR x=
x = (x cannot have a negative value.)
Q: Considering another infinite surd where the first term is , to repeat the entire process and to find the exact value for this surd.
ANSWER:
= = 1.8477...
= OR = = 1.9615...
= OR = = 1.9903...
As it is observed that = , therefore it can be understood that = as the trend has been so until now.
Hence,
→ = = 1.9975...
→ = = 1.9993...
→ = = 1.9998...
→ = = 1.9999...
→ = = 1.9999...
→ = = 1.9999...
→ = = 1.9999...
[NOTE: VALUES OBTAINED USING THE GRAPHICAL DISPLAY CALCULATOR: CASIO fx- 9860G]
From the above graph we realize as the value starts to increase after a certain number (1.999) it stabilizes and remains almost the same through out the graph. As we can see after 5 the value continues to remain the same till 10 and onwards. The general idea in both the question are that after the value starts to increase the graph stabilizes and the value remains the same.
Q: Find out the exact value for this surd ?
ANSWER:
→ =
As n becomes very large, - = 0 as n is .
= = x.
x =
On squaring both sides –
→ =
→ = 2+x
→ - x – 2 = 0
→
→
→
→
→ OR
= 2
Q: Now, considering the general infinite surd where the first term is , find an expression for the exact value of this general infinite surd in terms of k.
ANSWER:
The formula in terms of k → =
The value of an infinite surd is not always an integer.
Q: Find some values of k that make the expression an integer. Find the general statement that represents all the values of k for which the expression is an integer.
ANSWER:
=
= = x.
x =
On squaring both sides –
→ = k + x
→ - x – k = 0
→
→
OR
(x cannot be negative)
→ suppose:
- 1+4k=0
4k=0
k=0
- 1+4k=9
4k=8
K=2
- 1+4k= 49
4k=48
k= 12
Therefore, the sequence obtained is – 0, 2,6,12.... , n
= 0
= 2
= 6 → + 4= 2+4
= 12 → + 6 = 2+4+6
= 2+4+6+8
= 2+4+6+8+10+......
= {2 + (n-1) d}
→ = {2(2) + (n-1) 2}
→ = (4+2n-2)
→ = (2+2n)
= +
un = n+
Q: test the validity of your general statement using other values of k.
ANSWER:
- Suppose n=5
Then, on substituting 5 as the value of n in the general statement we get:-
= 5 + 5^2 = 30
k = 30
Now substituting the value of k in the equation - x – k=0 we get:
-x-30=0
→ -6x+5x -30 =0
→ x(x-6) +5(x-6) =0
→ (x+5) (x-6) =0
Thus, x=6 OR x=-5
Hence, proved.
- Supposing n =5
Then, on substituting 6 as the value of n in the general statement we get:-
= 6+6^2 = 42
k=42
Now substituting the value of k in the equation - x – k=0 we get:
- x – 42=0
→ -7x+6x -42 =0
→ x(x-7) +6(x-7) =0
→ (x+6) (x-7) =0
Thus, x=7 OR x=-6
Hence, proved.
HENCE, IT CAN BE CONCLUDED THAT EVEN AFTER USING OTHER VALUES OF “k” THE ANSWER OBTAINED IS AN INTEGER WHICH PROVES THE OBTAINED GENERAL STATEMENT VALID.
The scope of the general statement is that the need to always obtain an integer value as the answer is always satisfied by this general statement. The limitation of the general statement is that the use of a negative integer is invalid.
CONCLUSION:
By this I conclude that them I learnt how to use the surds and the correct time to use them.
