Question

Bilangan yang terdekat dengan bentuk:
2014^2-2013^2 2012^2-2011^2 ... 4^2-3^2 2^2-1^2
A. 5 x 10^5
B. 1 x 10^6
C. 2 x 10^6
D. 5 x 10^6
E. 2 x 10^7

Answer 1

2014² - 2013² + 2012² - 2011² + ........... 4² - 3² + 2² - 1²

= (2014 + 2013) (2014 - 2013) + (2012 + 2011) (2012 - 2011) + (2010 + 2009) (2010 - 2009) + ...... (4 + 3) (4 - 3) +  (2 + 1) (2 - 1)
= (4027 x 1) + (4023 x 1) + (4019 x 1) + ..... + (7 x 1) + (3 x 1)
= 4027 + 4023 + 4019 + ..... + 7 + 3

a = 3
b = 4
Un = 4027

* Mencari nilai n
Un = a + (n - 1)b
4027 = 3 + (n - 1)4
4027 - 3 = (n - 1)4
4024 = (n - 1)4
1006 = n - 1
n = 1007

* Mencari Sn
Sn = n/2 (a + Un)
S₁₀₀₇ = 1007/2 (3 + 4027)
          = 1007/2 (4030)
          =1007 x 2015
          = 2029105
          = 2 x 10⁶

Jadi, bentuk yang paling mendekati adalah 2 x 10⁶ (Jawaban C)

Answer 2

2014²-2013²+2012²-2011²+....+4²-3²+2²-1²
= (2014+2013)(2014-2013) + (2012+2011)(2012-2011) +...+(4+3)(4-3)+(2+1)(2-1)
= 4027 x 1 + 4023x1 + ... + 7x1 + 3 x 1
= 4027 + 4023 +... + 7 + 3

gunakan Sn barisan arit dgn a = 4027 dan b = -4 serta Un = 3
Un = a + (n-1)b
3  = 4027 + (n-1)-4
3  = 4027 - 4n + 4
4n = 4031 - 3
4n = 4028
  n = 4028 / 4
     = 1007

Sn = n / 2 (a+Un)
maka
2014²-2013²+2012²-2011²+....+4²-3²+2²-1²
= (2014+2013)(2014-2013) + (2012+2011)(2012-2011) +...+(4+3)(4-3)+(2+1)(2-1)
= 4027 x 1 + 4023x1 + ... + 7x1 + 3 x 1
= 4027 + 4023 +... + 7 + 3
= 1007/2 (4027+3)
= 1007/2 x 4030
= 1007 x 2015
= 2029105
= 2 x 10⁶