Common Mistakes
Traps in Sequence & Series
5 mistake patterns students fall for. Each one shows the wrong approach vs the correct approach.
Off-by-one in AP term numbering
FORMULAUsing a_n = a + nd instead of a + (n-1)d. The 1st term is a, not a+d.
✗ WRONG: 5th term = a + 5d
✓ RIGHT: 5th term = a + 4d (always subtract 1 from the term number)
Forgetting r=1 case in GP
DOMAINThe GP sum formula S_n = a(r^n-1)/(r-1) is undefined when r=1.
✗ WRONG: Applying the formula blindly and getting 0/0
✓ RIGHT: When r=1, all terms equal a, so S_n = na. Always check r=1 separately.
Infinite GP applied when |r| >= 1
DOMAINS = a/(1-r) only works when |r|<1. Series diverges otherwise.
✗ WRONG: Sum of 1+2+4+8+... = 1/(1-2) = -1
✓ RIGHT: This GP diverges since |r|=2>1. No finite sum exists.
Confusing S_n with a_n
FORMULAStudents sometimes use S_n formula when asked for a_n or vice versa.
✗ WRONG: Asked for the 10th term, computing S_10 instead
✓ RIGHT: a_n = S_n - S_{n-1}. The nth term is the difference of consecutive partial sums.
Sign error in method of differences
SIGN ERRORWhen computing T_n = S_n - S_{n-1}, students mess up the algebra.
✗ WRONG: S_n = n^2+n, so T_n = n^2+n - (n-1)^2-(n-1) = 2n (wrong sign)
✓ RIGHT: T_n = (n^2+n) - ((n-1)^2+(n-1)) = n^2+n-n^2+2n-1-n+1 = 2n