Arithmetic Sequences
An arithmetic sequence adds a constant difference $d$ between consecutive terms: $a, a+d, a+2d, \ldots$
Geometric Sequences
A geometric sequence multiplies each term by a constant ratio $r$: $a, ar, ar^2, ar^3, \ldots$
Sum Shortcuts for Consecutive Integers
| Sequence | Sum |
|---|---|
| 1 to $n$ | $\dfrac{n(n+1)}{2}$ |
| Odd integers 1 to $2n-1$ | $n^2$ |
| $a$ to $b$ (evenly spaced) | $\dfrac{(b-a+1)(a+b)}{2}$ |
| Count of integers from $a$ to $b$ | $b - a + 1$ |
GMAT Pattern Recognition
Units digits of powers cycle: $2^1=2, 2^2=4, 2^3=8, 2^4=16$ → cycle {2,4,8,6}. Use remainder when dividing by cycle length to find any term.
GMAT may define non-standard sequences recursively: $a_n = a_{n-1} + a_{n-2}$. Just follow the rule and build terms step by step.
10 Sequence Traps
1. nth term: it's $(n-1)d$, not $nd$
$a_n = a_1 + (n-1)d$. For the 5th term, use $d$ multiplied by 4, not 5.
2. Count of integers from $a$ to $b$: use $b-a+1$
From 3 to 9: $9-3+1 = 7$ integers. Many students say $9-3=6$.
3. Sum of arithmetic sequence: use mean × count
Mean = $(a_1+a_n)/2$. Sum = mean × number of terms.
4. Geometric ratio vs arithmetic difference
In geometric, you MULTIPLY. In arithmetic, you ADD. Don't mix them.
5. Careful with "every other" counting
Every 3rd integer from 1 to 100: count = $\lfloor 100/3 \rfloor = 33$ or $\lfloor (100-1)/3 \rfloor + 1$.
6. Cyclical patterns: divide index by cycle length
Units digit of $7^{53}$: cycle of 7 is {7,9,3,1} (length 4). $53 = 4 \times 13 + 1$. Remainder 1 → 1st position → 7.
7. Sum of first $n$ odd integers = $n^2$
This is a commonly tested fact: $1+3+5+\ldots+(2n-1)=n^2$.
8. Mean of arithmetic sequence = midpoint
For {4,7,10,13,16}: mean = $(4+16)/2 = 10$ = middle term. Always true for arithmetic sequences.
9. Recursive sequences: build step by step
Don't look for a shortcut in recursive sequences — just compute each term.
10. Geometric sequence: terms could alternate sign
If $r < 0$, terms alternate between positive and negative. Check sign behavior.
10 GMAT Practice Questions
The first term of an arithmetic sequence is 5 and the common difference is 3. What is the 20th term?
(C) 62. $a_{20} = 5 + (20-1)(3) = 5 + 57 = 62$.
What is the sum of all integers from 1 to 50?
(B) 1275. $S = \frac{50(51)}{2} = \frac{2550}{2} = 1275$.
In a geometric sequence, the first term is 2 and the common ratio is 3. What is the 5th term?
(C) 162. $a_5 = 2 \times 3^{5-1} = 2 \times 81 = 162$.
How many integers are there from 15 to 84, inclusive?
(C) 70. Count = $84 - 15 + 1 = 70$.
What is the sum of all even integers from 2 to 100?
(C) 2550. Even integers 2 to 100: 50 terms. Mean = $(2+100)/2 = 51$. Sum = $51 \times 50 = 2550$.
A sequence is defined by $a_1 = 3$ and $a_n = 2a_{n-1} - 1$ for $n > 1$. What is $a_4$?
(D) 15. $a_1=3$. $a_2=2(3)-1=5$. $a_3=2(5)-1=9$. $a_4=2(9)-1=17$. Wait: $a_4=17$, which is choice (E). (E) 17 is correct.
What is the units digit of $7^{42}$?
(A) 1. Units digits of $7^n$ cycle: 7, 9, 3, 1, 7, 9, 3, 1... (period 4). $42 = 4 \times 10 + 2$. Remainder 2 → 2nd position in cycle → 9. Wait, remainder 2 → second element = 9. (D) 9 is correct.
What is the sum of the first $n$ positive odd integers?
(A) $n^2$. Sum of first $n$ odd integers: $1+3+5+\ldots+(2n-1)=n^2$. This is a classic identity.
Is a sequence arithmetic?
(1) The difference between any two consecutive terms is 7.
(2) The sum of any three consecutive terms is divisible by 3.
(A) Statement (1) alone sufficient. (1): Constant difference of 7 between consecutive terms — this IS the definition of arithmetic. Sufficient. (2): This is true for arithmetic sequences but also for other sequences. Not sufficient alone. Answer: (A).
A geometric sequence has first term 5 and third term 20. What is the second term?
(B) 10. $a_3 = a_1 r^2$ → $20 = 5r^2$ → $r^2 = 4$ → $r = 2$. $a_2 = 5 \times 2 = 10$.
Lesson Summary — Key Takeaways
Arithmetic nth term: a₁ + (n−1)d
The 1st term plus $(n-1)$ times the common difference.
Count from a to b: b−a+1
3 to 9 inclusive = 9−3+1 = 7. The "+1" is always needed.
Sum = mean × count (arithmetic)
Mean of arithmetic sequence = (first + last)/2. Multiply by count for sum.
Cyclical units digits: divide by cycle length
Powers of 2 cycle {2,4,8,6}. Find remainder of exponent ÷ 4 to identify position.