Bunuel wrote:
For a sequence \(t_1\), \(t_2\), \(t_3\), ..., \(S_n\) denotes the sum of the first n terms of a sequence. If \(S_n = n^3 + n^2 + n + 1\), and \(t_m = 291\), then m is equal to?
A. 10
B. 22
C. 24
D. 26
E. 30
Are You Up For the Challenge: 700 Level QuestionsIf I look at the question from GMAT perspective and solve it within the gambit of what is asked in GMAT, then I will look for a pattern.
Now
\(S_n = n^3 + n^2 + n + 1\)
1) When n=1, \(S_n = n^3 + n^2 + n + 1\) => \(S_1 = 1^3 + 1^2 + 1 + 1=4\)....\(S_1=t_1=4\)
2) When n=2, \(S_n = n^3 + n^2 + n + 1\) => \(S_2 = 2^3 + 2^2 + 2 + 1=15\)....\(S_2=t_1+t_2=4+t_2=15....t_2=11\)
3) When n=3, \(S_n = n^3 + n^2 + n + 1\) => \(S_3 = 3^3 + 3^2 + 3 + 1=40\)....\(S_3=S_2+t_3=15+t_3=40....t_3=25\)
4) Similarly \(t_4=45........ t_5=71........ t_6=103\)
Sequence is 4, 11, 25, 45, 71, 103.....
We can search either for next 3-4 values as 291 is not a very big value to look for OR find a pattern in the sequence..
So, let us look for the sequence..
The sequence is 4, 11, 25, 45, 71, 103.....
Difference in consecutive numbers is 7, 14, 20, 26, 32, That if after first term, the difference increases by 6..
Thus the sequence becomes 4, 11, 25, 45, 71, 103, 103+(32+6)=141, (141)+(38+6)=185, 185+(44+6)=235, 235+(50+6)=291(our term)
so sequence is 4, 11, 25, 45, 71, 103, 141, 185, 235, 291, and 291 is the \(10^{th}\) term.
A