2

Part of 2012 AIME Problems

Problems(2)

An arithmetic sequence with terms increased

Source: 2012 AIME I Problem 2

The terms of an arithmetic sequence add to

715

. The first term of the sequence is increased by

1

, the second term is increased by

3

, the third term is increased by

5

, and in general, the

k

th term is increased by the

k

th odd positive integer. The terms of the new sequence add to

836

. Find the sum of the first, last, and middle terms of the original sequence.

arithmetic sequenceAMC

Similar Geometric Sequences

Source: 2012 AIME II Problem 2

Two geometric sequences

a_1,a_2,a_3,\ldots

b_1,b_2,b_3\ldots

have the same common ratio, with

a_1=27

b_1=99

a_{15}=b_{11}

a_9.