Q01 |
- Difference in linear function sequences
Q02 |
- Difference in quadratic function sequences : y=a*x^2+b*x+c
Q03 |
- Difference in cubic function sequences : y=a*x^3+b*x^2+c*x+d
Q04 |
- Find function of y in terms of x for a given sequences
Q05 |
- More than 50 series can be found
Q06 |
- Special series on internet
Q07 |
- Find the sum of 1/(1*2) + 1/(2*3) + 1/(3+4) + .... + 1/(999*1000)
Q08 |
- Three numbers in arithmetic sequence.
Q09 |
- Formula
Q10 |
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Answers
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Q01. Difference in linear function sequences : y = a*x + b
Solution
* ..... x ....... y ....... y'
* ..... 1 ..... a+b ....... a
* ..... 2 .... 2a+b ....... a
* ..... 3 .... 3a+b ....... a
* ..... 4 .... 4a+b ....... a
* ..... 5 .... 5a+b ....... a
First derivative y' is the first difference = a
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Q02. Difference in quadratic function sequences : y=a*x^2+b*x+c
Solution
* x ......... y ...... y' ..... y"
* 0 ......... c
* 1 ..... a+b+c .... a+b
* 2 ... 4a+2b+c ... 3a+b ..... 2a
* 3 ... 9a+3b+c ... 5a+b ..... 2a
* 4 .. 16a+4b+c ... 7a+b ..... 2a
Second derivative y" is the second difference = 2a
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Q03. Difference in cubic function sequences : y=a*x^3+b*x^2+c*x+d
Find the difference
* x ............. y ......... y' ..... y" .... 3rd diff
* 1 ....... a+b+c+d ..... a+b+c
* 2 .... 8a+4b+2c+d ... 7a+3b+c ... 6a+2b
* 3 ... 27a+9b+3c+d .. 19a+5b+c .. 12a+2b .... 6a
* 4 .. 64a+16b+4c+d .. 37a+7b+c .. 18a+2b .... 6a
* 5 . 125a+25b+5c+d .. 61a+9b+c .. 24a+2b .... 6a
The 3rd difference = 6a (Note : 3rd derivatve = 6a)
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Q04. Find function of y in terms of x for the following sequences
- x = 0, 1, 2, 3, 04, 05, 06, 07, 08, ....
- y = 0, 1, 3, 6, 10, 15, 21, 28, 36, ....
Solution
[Method 1] By finding dfiiference
* y = 03, 06, 10, 15, 21, 28, 35, ........
* y'= --, 03, 04, 05, 06, 07, 08, ........
* y"= --, --, 01, 01, 01, 01, 01, ........
- Let the function be y=a*x^2+b*x+c
- Because 2nd diff = 2a =1. Hence a=1/2.
- Also we can see y=0 as x=0. Thus c = 0.
- Now we have y=x^2/2 + b*x.
- We know that x=1 and y=1 and we get b=1/2.
- The requred function y = x(x+1)/2.
- This is number in triangular pattern
[Method 2] By observation
- T(0) = 0
- T(1) = 1
- T(2) = 1 + 2 = 3
- T(3) = 1 + 2 + 3 = 6
- T(4) = 1 + 2 + 3 + 4 = 10
- T(n) = 1 + 2 + 3 + 4 + ...... + n = n*(n+1)/2
- Hence the functions is y = x*(x+1)/2
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Q05. More than 50 series can be found
Examples
a. Arithematic Progression (A.P.) : See keyword Arithematic series
b. Geometric Progression (G.P.) : See keyword Geometric series
c. Harmonci Progression (H.P.) : See keywordHarmonic series
d. P series
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Q06. Special series on internet
C.
On Internet Sum[n*(n+1)/2] = n*(n+1)*(n+2)/3!
D.
On Internet Pascal triangle and series
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Q07. Find the sum of 1/(1*2) + 1/(2*3) + 1/(3+4) + .... + 1/(999*1000)
Solution
- 1/(1*2) = 1/1 - 1/2
- 1/(2*3) = 1/2 - 1/3
- 1/(3*4) = 1/3 - 1/4
- .......
- 1/(998*999) = 1/998 - 1/999
- 1/(999*1000)= 1/999 - 1/1000
- Sum left above lines = 1/(1*2) + 1/(2*3) + 1/(3+4) + .... + 1/(999*1000)
- Sum right above lines = 1 - 1/1000 = 0.999
Answer
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Q08. Three numbers in arithmetic sequence. if 1st number adds 9, 2nd number adds 7
and 3rd number adds 9 and these new numbers will be in geometric sequence.
Find these numbers in postive values.
Solution
- Let the three numbers be x-d, x, x+d
- Then ((x-d)+9)/(x+7) = (x+7)/((x+d)+9)
- Hence ((x-d)+9)*((x+d)+9) = (x+7)^2
- Simplify we have 4*x = d^2 - 32
- If d = 8 and x = 8 then numbers are 00, 08, 16
- If d =10 and x =17 then numbers are 07, 17, 27
- If d =12 and x =24 then numbers are 12, 24, 36
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Q09. Formula
- Sum[1] = n if there are n terms
- Sum[n] = n*(n+1)/2
- Sum[n^2] = n*(n+1)*(2*n+1)/6
- Sum[n^3] = (n*(n+1)/2)^2
- Sum[n*(n+1)/2] = n*(n+1)*(n+2)/3!
- Arithmetic Progression (A. P.)
- T(1) = a is the first term
- T(n)=a+(n-1)*d where d is common difference = T(n)-T(n-1)
- S(n) = n*(T(1)+T(n))/2
- a,b,c in arithmetric sequence then b-a = c-b and b is arithmetric mean
- Geometirc progression
- T(1) = a
- T(n) = a*r^(n-1) where r is common ration = T(n)/T(n+1)
- S(n) = a*(1-r^n)/(1-r)
- If r < 1 then S(n) = a/(1-r) and the series is convergent
- if r >= 1 then sequence is diverge
- a,b,c in geometric sequence the a/b = b/c and b is geometric mean
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Q10. Answer
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