2008 iTest Problems/Problem 43

Problem

Alexis notices Joshua working with Dr. Lisi and decides to join in on the fun. Dr. Lisi challenges her to compute the sum of all $2008$ terms in the sequence. Alexis thinks about the problem and remembers a story one of her teachers at school taught her about how a young Karl Gauss quickly computed the sum $1+2+3+\cdots+98+99+100$ in elementary school. Using Gauss's method, Alexis correctly finds the sum of the $2008$ terms in Dr. Lisi's sequence. What is this sum?

Note: Dr. Lisi’s sequence is $-1776, -1765, -1754 \cdots$

Solution

The first term is $-1776$ and the common difference is $11$, so the $2008^\text{th}$ term is $-1776 + 11 \cdot 2007$. Using the arithmetic series formula, the sum of the $2008$ terms in Dr. Lisi's sequence is $\tfrac{2008(-1776-1776+11\cdot 2007)}{2} = \boxed{18599100}$.

See Also

2008 iTest (Problems)
Preceded by:
Problem 42
Followed by:
Problem 44
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100
Invalid username
Login to AoPS