Urgent! use gauss's approach to find the following sums 1+3+5+7+ formula and solve

The sum of the first 499 terms is 249001

Step-by-step explanation:

* Lets revise the arithmetic sequence

- There is a constant difference between each two consecutive

  numbers

- Ex:

# 2  ,  5  ,  8  ,  11  ,  ……………………….

# 5  ,  10  ,  15  ,  20  ,  …………………………

# 12  ,  10  ,  8  ,  6  ,  ……………………………

* General term (nth term) of an Arithmetic sequence:

- U1 = a  ,  U2  = a + d  ,  U3  = a + 2d  ,  U4 = a + 3d  ,  U5 = a + 4d

- Un = a + (n – 1)d, where a is the first term , d is the difference

 between each two consecutive terms n is the position of the

 number

- The sum of first n terms of an Arithmetic sequence is calculate from

 Sn = n/2[a + l], where a is the first term and l is the last term

* Now lets solve the problem

∵ The terms of the sequence are 1 , 3 , 5 , 7 , , 997

∵ The first term is 1 and the second term is 3

∴ The common difference d = 3 - 1 = 2

∵ The first term is 1

∵ The last term is 997

∵ The common difference is 2

- Lets find how many terms in the sequence

∵ an = a + (n - 1) d

∴ 997 = 1 + (n - 1) 2 ⇒ subtract 1 from both sides

∴ 996 = (n - 1) 2 ⇒ divide both sides by 2

∴ 498 = n - 1 ⇒ add 1 for both sides

∴ n = 499

∴ The sequence has 499 terms

- Lets find the sum of the first 499 terms  

∵ Sn = n/2[a + l]

∵ n = 499 , a = 1 , l = 997

∴ S499 = 499/2[1 + 997] = 499/2 × 998 = 249001

* The sum of the first 499 terms is 249001

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