AIME 2012 I · 第 2 题

Solutions

Solution 1

If the sum of the original sequence is $\sum_{i=1}^{n} a_i$ then the sum of the new sequence can be expressed as $\sum_{i=1}^{n} a_i + (2i - 1) = n^2 + \sum_{i=1}^{n} a_i.$ Therefore, $836 = n^2 + 715 \rightarrow n=11.$ Now the middle term of the original sequence is simply the average of all the terms, or $\frac{715}{11} = 65,$ and the first and last terms average to this middle term, so the desired sum is simply three times the middle term, or $\boxed{195}.$

Alternatively, notice that in the original sequence, $11a_1 + 55d = 715$, from which $a_1 + 5d = 65$. Since we are tasked to find $a_1 + a_6 + a_{11} = 3(a_1 + 5d)$, the desired answer is $3 \cdot 65 = \boxed{195}.$

Solution 2

After the adding of the odd numbers, the total of the sequence increases by $836 - 715 = 121 = 11^2$. Since the sum of the first $n$ positive odd numbers is $n^2$, there must be $11$ terms in the sequence, so the mean of the sequence is $\dfrac{715}{11} = 65$. Since the first, last, and middle terms are centered around the mean, our final answer is $65 \cdot 3 = \boxed{195}$

Solution 3

Proceed as in Solution 2 until it is noted that there are 11 terms in the sequence. Since the sum of the terms of the original arithmetic sequence is 715, we note that $\frac{2a_1 + 10d}{2} \cdot 11 = 715$ or $2a_1 + 10d = 130$ for all sets of first terms and common differences that fit the conditions given in the problem. Assume WLOG that $a_1 = 60$ and $d = 1$. Then the first term of the corresponding arithmetic sequence will be $60$, the sixth (middle) term will be $65$, and the eleventh (largest) term will be $70$. Thus, our final answer is $60 + 65 + 70 = \boxed{195}$.

~ cxsmi

Solution 4 (Similar to the above solutions)

Let the sequence be $a,a+d,a+2d,..., a+(k-1)d$. Since there are $k$ terms, we have an equation $k^2 = 836-715 = 121$. Solving, we get $k$ as $11$. Replacing $k$ in our sequence with $11$, we get $a,a+d,a+2d,..., a+10d$. The sum of this sequence is equal to $715$, or $\frac{(2a+10d)(11)}{2} = 715$, and by simplifying we get $a+5d=65$. We are asked for the first, middle, and end terms, which are $a$,$a+5d$, and $a+10d$ respectively. Their sum is $3a+15$, or $3 \cdot (a+5d)$. Our desired answer is $65 \cdot 3 = \boxed{195}$.

~Irfans123

Video Solution by Richard Rusczyk

https://artofproblemsolving.com/videos/amc/2012aimei/298

~ dolphin7

Video Solution

https://www.youtube.com/watch?v=T8Ox412AkZc ~Shreyas S

Video Solution by OmegaLearn

https://youtu.be/tKsYSBdeVuw?t=4689

~ pi_is_3.14