Linear programming, Skinner's Programming, Straight line programming, Model for linear programming, Linear programming on the topic Arithmetic Sequences

1. (Skinner’s Linear Program)
Topic: Arithmetic Sequences

2. ARITHMETIC SEQUENCES

3. Frame: 1
In mathematics, an Arithmetic Progression(AP) or Arithmetic
Sequence is a sequence of numbers such that the difference
between the consecutive terms is a constant. For example,
2,4,6,8,… is an AP.
Check whether 1,3,5,7,… is an AP or not?

4. Frame: 2
Frame-1 Answer: Yes.
The initial term of an arithmetic sequence is called the First -
term of the arithmetic sequence.
Write the first term of the arithmetic sequence 4,7,10,13,…

5. Frame-2 Answer: 4
The constant difference between two consecutive terms of an
arithmetic sequence is called the Common Difference.
Find the common difference of the arithmetic sequence 3,5,7,…
Frame: 3

6. Frame: 4
Frame-3 Answer: 2
The arithmetic sequence which have a first term and a last
term, and all the terms follow a specific order is called Finite
Arithmetic Sequence. Otherwise, the sequence is called Infinite
Arithmetic Sequence. For example, ‘set of all natural numbers
below 100’ is an finite arithmetic sequence.
Is the ‘set of all even natural numbers’ is an example for finite
arithmetic sequence?

7. Frame: 5
Frame-4 Answer: No.
The set of all even natural numbers has infinite terms. Hence
it is an example for infinite arithmetic sequence.
Is the ‘set of all odd natural numbers’ is an example for
infinite arithmetic sequence?

8. Frame: 6
Frame-5 Answer: Yes.
For any arithmetic sequence, the sum of three consecutive terms
is thrice the middle one. That is, x+y+z=3y.
Eg:- 1,3,5,7,… is an AP. 3+5+7=15=3×5.
15,20,25,30…is an AP. Find the sum of first three terms of
this AP?

9. Frame: 7
Frame-6 Answer: 60 (15+20+25=3×20=60)
If we take the first term of an AP as f and the common difference as d,
then the nth term is, f+(n-1)d.
Eg.1:- 9 th terms of the AP 4,7,10,… is;
4+(9-1)×3=28. (Here, f=4 and d=3)
Eg.2:- 11 th term of the AP 3,5,7,… is;
3+(11-1)×2=23. (Here, f=3 and d=2)
Find the 21st term of the AP 8,10,12,14,…?

10. Frame-7 Answer: 48. [8+(21-1)×2= 48]
The sum of any number of consecutive natural numbers, starting
with one, is half the product of the last number and the next natural
number. That is; 1+2+3+….+n = ½ n(n+1).
Eg.1:- 1+2+3+…+10 = ½ [10×(10+1)]= ½ [10×11]= 5×11=55
Eg.2:- 1+2+3+…+50 = ½ [50×(50+1)]= ½ [50×51]= ½[2550]=1275
Frame: 8
Find the sum of first 20 natural numbers.

11. Frame: 9
Frame-8 Answer: 210 (1+2+3+…+20=½[20×(20+1)]=10×21=210)
The sum of first n terms of an AP= n/2 [2a+(n-1)d]; where a and
d are first term and common difference respectively.
Eg.1:- Sum of first 20 terms of the AP 1,4,7,… is;
Sum= 20/2 [(2×1)+(20-1)×3]= 10×[2+(19×3)]= 590
Eg.1:- Sum of first 32 terms of the AP 3,5,7,… is;
Sum= 32/2 [(2×3)+(32-1)×2]= 16×[6+(31×2)]= 1088.
Find the sum of first 10 terms of the AP 2,5,8,…

12. Frame:10
Frame-9 Answer:155.
[Sum=10/2[(2×2)+(10-1)×3]= 5×[4+(9×3)]= 155.]

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Shahaziya.A.P