Max@Math Revolution wrote:[Math Revolution GMAT math practice question]

The sequence Sn has the terms S1=1, S2=2, S3=6, S4=15, S5=31, ....... If n is a positive integer, what could be the value of Sn-Sn-1 in terms of n?

A. n
B. n^2
C. -n
D. -n^2
E. (n-1)^2

\[?\,\,\,:\,\,\,S{\,_N} - {S_{N - 1}} = f\left( N \right)\]
One possible PATTERN becomes evident below:
$${S_2} - {S_{\boxed1}} = 2 - 1 = 1 = {\boxed1^2}$$
$${S_3} - {S_{\boxed2}} = 6 - 2 = 4 = {\boxed2^2}$$
$${S_4} - {S_{\boxed3}} = 15 - 6 = 9 = {\boxed3^2}$$
$${S_5} - {S_{\boxed4}} = 31 - 15 = 16 = {\boxed4^2}$$
$$ \ldots $$
\[? \,\, : \,\, {S_N} - {S_{\boxed{N - 1}}} = {\boxed{N - 1}^{\,2}}\]

Hence we are sure (E) is viable (we found a "logic rule" that obeys the given first terms)! (*)


This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.

(*) P.S.: each of the other alternative choices does not obey (all) the given first terms. This guarantees the question stem is ok!

=>

We begin by writing out the terms of the sequence:

S2-S1 = 2 -1 = 1=1^2
S3-S2 = 6 - 2 = 4 = 2^2
S4-S3 = 15 - 6 = 9 = 3^2
S5-S4 = 31-15 = 16 = 4^2
...
Following the above pattern gives
Sn-Sn-1 = (n-1)^2

Therefore, the answer is E.
Answer: E