Even integers

Algebra Level 2

The sum of 4139 4139 consecutive positive even integers is 17135460 17135460 . Find the sum of the lowest and highest integers.


The answer is 8280.

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1 solution

The sequence of numbers forms an arithmetic progression. The common difference is 2 2 ; number of terms is 4139 4139 ; and the sum is of the terms is 17135460 17135460 . We can use the formula:

s = n 2 ( a 1 + a n ) s=\dfrac{n}{2}(a_1+a_n) where: s = s u m s=sum , n = n u m b e r o f t e r m s n=number~of~terms , a 1 = f i r s t t e r m a_1=first~term and a n = n t h t e r m a_n=n^{th}~term

Substituting, we get

17135460 = 4139 2 ( a 1 + a 4139 ) 17135460=\dfrac{4139}{2}(a_1+a_{4139})

17135460 ( 2 ) 4139 = a 1 + a 4139 \dfrac{17135460(2)}{4139}=a_1+a_{4139}

a 1 + a 4139 = a_1+a_{4139}= 8280 \boxed{8280} answer \color{#D61F06}\boxed{\text{answer}}

Bonus: What do you think might be the sum a 1 + k + a 4139 k a_{1+k} + a_{4139-k} where ( k N , 0 k 2069 ) \left( k \in \mathbb{N}, \ 0 \le k \le 2069 \right) . Does it depend on the value of k k ?

Tapas Mazumdar - 4 years, 1 month ago

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