Expect 2-3 questions from sequences and series on the GMAT. The questions are easy and test you only on basic concepts. Learn the formulae well and their application. We will learn Arithmetic, Geometric and Harmonic progressions and series. Read on.

Aritmetic Progression

An arithmetic progression is a sequence of numbers where each new term after the first is formed by adding a fixed amount called the common difference to the previous term in the sequence. For example the sequence 3,5,7,9,11,...
The common difference can be negative: for example the sequence 2,-1,-4,-7,... is an arithmetic progression with first term 2 and common difference -3.

In general we can write an arithmetic progression as follows:

a, a + d, a + 2d, a + 3d, ...

Tn = a + (n - 1) d

and in general

T = a + (m - n ) d n m

The sum of the members of a finite arithmetic progression is called an arithmetic series.

Express the arithmetic series in two different ways:

S = a + (a +d )+ (a +2d )+ ⋅⋅⋅+ (a + (n - 3)d )+ (a + (n - 2)d )+ (a + (n - 1)d ) n 1 1 1 1 1 1

Sn = (a1+ (n - 1)d )+ (a1+ (n - 2)d )+ (a1+ (n - 3)d )⋅⋅⋅+ (a1+2d )+ (a1+d )+a1

Add both sides of the two equations. All terms involving d cancel, and so we’re left with:

2S = n (2a + (n - 1 )d ) n 1

Rearranging and remembering that an = a1 + (n - 1 ) d, we get:

n [2a1 + (n - 1 )d] n (a1 + an ) Sn = ------------------------= --------------- 2 2

Sometimes it is useful to take the terms in an AP as

..., a - 2d, a - d, a, a + d, a + 2d, ...

when it’s given that the number of terms is odd.

If the number of terms is even you should use this form, if applicable:

..., a - 3d, a - d, a + d, a + 3d, ...

where 2d is the common difference.

Geometric Progression

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed number called the common ratio. The sequence 1, 3, 9, 27, . . .is a geometric progression with first term 1 and common ratio 3. The common ratio could be a fraction and it might be negative. For example, the geometric progression with first term 2 and common ratio 1 - -- 3

is : 2- 2- -2-- -2-- 2, - , , - , , ... 3 9 27 81

In general we can write a geometric progression as follows:

2 3 4 a, ar, ar , ar , ar , ...

where the first term is a sand the common ratio is r

Some important results concerning geometric progressions:

The nth term of a GP is given by n - 1 Tn = ar

A geometric series is the sum of the numbers in a geometric progression:

n ∑ k 0 1 2 3 n ar = ar + ar + ar + ar + ⋅ ⋅ ⋅ + ar k=0

We can find a simpler formula for this sum by multiplying both sides of the above equation by (1 - ), and we’ll see that

n ∑ k n+1 (1 - r) ar = a - ar k=0

Since all the other terms cancel. Rearranging (for r ⁄= 1) gives the convenient formula for a geometric series:

n+1 ∑n k a-(r---------1)- ar = k=0 r - 1

Note: If one were to begin the sum not from 0, but from a higher term, say m, then

∑n k a(rn+1 - rm ) ar = ------------------ k=m r - 1

If the common ratio in a geometric series is less than 1 in modulus, (that is -1<r<1), the sum of an infinite number of terms can be found. This is known as the sum to infinity, S ∞, where

---a---- S ∞ = 1 - r

The behaviour of a geometric sequence depends on the value of the common ratio.
If the common ratio is:

  1. Positive, the terms will all be the same sign as the initial term.
  2. Negative, the terms will alternate between positive and negative.
  3. Greater than 1, there will be exponential growth towards positive infinity.
  4. 1, the progression is a constant sequence.
  5. Between -1 and 1 but not zero, there will be exponential decay towards zero.
  6. -1, the progression is an alternating sequence
  7. Less than -1, for the absolute values there is exponential growth towards infinity.

Harmonic Progression

A series of quantities are said to be in Harmonic progression when their reciprocals are in Arithmetic Progression.

e.g. 1/3, 1/5, 1/7, 1/9,…

1 Tn = ------------------- [a + (n - 1 )d]

Means

Aritmetic Mean (AM) = a + b ------- 2

Geometric Mean (GM) = √ ---- ab

Harmonic Mean (HM) = 2ab ------- a + b

Also note that :

√ --------------- GM = AM ⋅ H M

AM > GM > H M

(You can remember this by AGH are in alphatecial order:A comes before G which comes before H)

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