Without computing each sum of the arithmetic sequence, find which is greater, O

Question

Without computing each sum of the arithmetic sequence, find which is greater, O or E, and by how much. O = 1 + 3 + 5 + 7 + … + 97 E = 2 + 4 (Billstein 13)
Billstein, Rick. Problem Solving Approach to Mathematics for Elementary School Teachers, A, 12th Edition. Pearson, 20160401. VitalBook file. Without computing each sum of the arithmetic sequence, find which is greater, O or E, and by how much. O = 1 + 3 + 5 + 7 + … + 97 E = 2 + 4 (Billstein 13)
Billstein, Rick. Problem Solving Approach to Mathematics for Elementary School Teachers, A, 12th Edition. Pearson, 20160401. VitalBook file. Without computing each sum of the arithmetic sequence, find which is greater, O or E, and by how much. O = 1 + 3 + 5 + 7 + … + 97 E = 2 + 4 (Billstein 13)
Billstein, Rick. Problem Solving Approach to Mathematics for Elementary School Teachers, A, 12th Edition. Pearson, 20160401. VitalBook file.

Explanation / Answer

O is a sum of arithmatic progressions in which first term is 1 and common difference is 2

Number of terms ={(97-1)÷2} +1 = 48 + 1 = 49

So Sum = 49/2 ({2×1 + (49-1)× 2} = 49/2 ( 2+ 96)

= 49/2 × 98 =. 49 × 49 = 2401

And E = 2+4 = 6

So O is greater and quantity = 2401 - 6 = 2395

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