54-04 Basics of Arithmetic & Geometric Progression

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Examples of Summation.

A) Find the sum of the first n terms of an AP 7, 3, -1, -5, ---.

Solution:
1) Here, a = 7, d = 3 - 7 = - 4 , so d = - 4 
[ Note: Let us check whether the given sequence is an AP or not Simply by calculating d for the next pair. i.e. d = - 1 - 3 = - 4. which is the same as the previous d so it is an AP ]
2) Now we can use the formula 

        S_n = n * [ 2 a + (n - 1) d ) ] / 2
        S_n = n * [ 2 * (7)+ (n - 1) * (- 4 ) ] / 2
        S_n = n * [ 14 + (- 4 n + 4) ] / 2
        S_n = n * [ 14 - 4 n + 4) ] / 2
        S_n = n * [ 18 - 4 n ) ] / 2
        S_n = n * [ 18/2 - 4 n /2 ]
        S_n = n * [ 9 - 2 n ]
3) Answer: So the sum of the first n terms of an AP is  S_n = n * [ 9 - 2 n ] 

B) Find the sum of all the terms of an AP  2, 5, 8, 11,---, 299.

Solution:
1) Here, a = 2, d = 5 - 2 = 3, so d = 3 and last term l = t_n = 299.
2) To find n, we need to use the formula, 
             t_n = a + ( n - 1 ) d
          299  = 2 + ( n - 1 ) * ( 3 )
          297  = ( n - 1 ) * ( 3 )
          297 / 3 = ( n - 1 )
          99  = ( n - 1 )
          99 + 1 = n
          n = 100
3) Now we can use the formula,       S_n = n * ( a + l ) / 2
          S_n = n * ( a + l ) / 2
          S_n = 100 * ( 2 + 299 ) / 2
          S_n = 100 * ( 301 ) / 2
          S_n = 50 * ( 301 )
          S_n = 15050
4) Answer: The sum of the terms of an AP  2, 5, 8, 11, ---, 299, is S_n = 15050.

Note: The Above problem in a different way:

C) Find the sum of the first 100 terms of an AP, 2, 5, 8, 11, ---.

Solution:
1) Here, a = 2, d = 5 - 2 = 3, so d = 3 and n = 100.

2) Here we can use the formula S_n = n * [ 2 a + (n-1) d ) ] / 2
          S_n = n * [ 2 a + (n-1) d ) ] / 2
          S_n = 100 * [ 2 (2) + (100-1) (3) ) ] / 2
          S_n = 50 * [ 4 + (99) (3) ) ]
          S_n = 50 * [ 4 + 297 ]
          S_n = 50 * [ 301 ]
          S_n = 15050
3) Answer: The sum of the first 100 terms of an AP  2, 5, 8, 11, ---, is S_n = 15050.

D) Find the sum of the first n natural numbers.

Solution:
1) For the natural numbers, a = 1, d = 1.
2) Here we can use the formula S_n = n * [ 2 a + (n-1) d ) ] / 2
          S_n = n * [ 2 (1) + (n-1) (1) ) ] / 2
          S_n = n * [ 2 + n - 1 ] / 2
          S_n = n * [ n + 1 ] / 2
3) Answer: The sum of the first n natural numbers, is S_n = n * [ n + 1 ] / 2.

In the next part, we will see a few examples and some essential formulae.

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Anil Satpute