**Summary I**

► By a __sequence__, we mean an arrangement of number in definite order according to some mile. Also, we define a sequence as a function whose domain is the set of natural numbers or some subsets of the type {1, 2, 3, …, *k*}. A sequence containing a finite number of terms is called a __finite sequence__. A sequence is called __infinite__ if it is not a finite sequence.

► Let *a*_{1}, *a*_{2}, *a*_{3}, … be the sequence, then the sum expressed as *a*_{1}+*a*_{2}+*a*_{3}+⋯ is called series. A series is called __finite series__ if it has got finite number of terms.

► An arithmetic progression (A.P.) is a sequence in which terms increase or decrease regularly by the same constant. This constant is called __common difference__ of the A.P. Usually, we denote the first term of A.P. by *a*, the common difference by *d* and the last term by ℓ. The __general term__ or the *n*th term of the A.P. is given by *a _{n}*=

*a*+(

*n*-1)

*d*.

The sum

*S*of the first

_{n}*n*terms of an A.P. is given by

*S*=½

_{n}*n*[2

*a*+(

*n*-1)

*d*]=½

*n*(

*a*+

*ℓ*)

► The __arithmetic mean__ *A* of any two numbers *a* and *b* is given by ½*n*(*a*+*b*). Thus, the sequence *a*, *A*, *b* is an arithmetic sequence.

► A sequence is said to be a __geometric progression__ or GP., if the ratio of any term to its preceding term is same throughout. This constant factor is called the __common ratio__. Usually, we denote the first term of a GP. by *a* and its common ratio by *r*. The general or the *n*th term of G.P. is given by *a _{n}*=

*a⋅r*

^{n-1}.

The sum

*S*of the first

_{n}*n*terms of GP. is given by

► The geometric mean (G.M.) of any two positive numbers

*a*and

*b*is given by √

*a⋅b*. Thus, the sequence

*a*,

*G*,

*b*is a G.P.

**Summary II**

In this chapter, you have studied the following points:

1. An **arithmetic progression** (AP) is a list of numbers in which each term is obtained by adding a fixed number *d* to the preceding term, except the first term. The fixed number *d* is called the **common difference**.

The general form of an AP is *a*, *a*+*d*, *a*+2*d*, *a*+3*d*, …

2. A given list of numbers *a*_{1}, *a*_{2}, *a*_{3}, … is an AP, if the differences *a*_{2}–*a*_{1}, *a*_{3}–*a*_{2}, *a*_{4}–*a*_{3}, …, give the same value, i.e., if *a*_{k+1}–*a _{k}* is the same for different values of k.

3. In an AP with first term

*a*and common difference

*d*, the

*n*th term (or the general term) is given by

*a*=

_{n}*a*+(

*n*-1)

*d*.

4. The sum of the first

*n*terms of an AP is given by:

*S*=½

_{n}*n*[2

*a*+(

*n*-1)

*d*]

5. If

*ℓ*is the last term of the finite AP, say the

*n*th term, then the sum of all terms of the AP is given by:

*S*=½

_{n}*n*(

*a*+

*ℓ*)

A Note To The Reader

Ifa,b,care in AP, thenb=½(a+c) andbis called the arithmetic mean ofaandc.

**Summary III**

1. *n*th term of General Term of an Arithmetic progression.

*a*=

_{n}*a*+(

*n*-1)

*d*

2. Arithmetic means between

*a*and

*b*

*A*=½(

*a*+

*b*)

3. Sum of the First

*n*terms of an arithmetic series.

(i) *S _{n}*=½

*n*[2

*a*+(

*n*-1)

*d*]

(ii) *S _{n}*=½

*n*(

*a*+

*ℓ*) when last term ‘ℓ’ is known.

4. General or *n*th term of a G.P

*a*=

_{n}*a⋅r*

^{n-1}

5. Geometric means between

*a*and

*b*

*G*=±√

*a⋅b*

6. Sum of

*n*terms of a Geometric Series

7. Sum of an infinite Geometric Series

**STUDY GUIDE**

**Concept Summary**

Big Ideas | Applying the Big Ideas |
---|---|

▪ An arithmetic sequence is related to a linear function and is created by repeatedly adding a constant to an initial number. An arithmetic series is the sum of the terms of an arithmetic sequence. | This means that: ▪ The common difference of an arithmetic sequence is equal to the slope of the line through the points of its related linear function. ▪ Rules can be derived to determine the nth term of an arithmetic sequence and the sum of the first n terms of an arithmetic series. |

▪ A geometric sequence is created by repeatedly multiplying an initial number by a constant. A geometric series is the sum of the terms of a geometric sequence. | ▪ The common ratio of a geometric sequence can be determined by dividing any term after the first term by the preceding term. ▪ Rules can be derived to determine the nth term of a geometric sequence and the sum of the first n terms of a geometric series. |

▪ Any finite series has a sum, but an infinite geometric series may or may not have a sum. | ▪ The common ratio determines whether an infinite series has a finite sum. |

**Skills Summary**

Skill | Description | Example |
---|---|---|

Determine the general term, t, for an arithmetic sequence._{n}(1.1, 1.2) |
A rule is:t=_{n}t_{1}+d(n-1)where t_{1} is the first term, d is the common difference, and n is the number of terms. |
For this arithmetic sequence: -9, -3, 3, 9, … the 20th term is: t_{20}=-9+6(20-1)t_{20}=-9+6⋅(19)t_{20}=105 |

Determine the sum of n terms, S, for an arithmetic series. (1.2)_{n} |
When n is the number of terms, t_{1} is the first term, t is the _{n}nth term, and d is the common differenceOne rule is: S=½_{n}n(t_{1}+t)_{n}Another rule is: S=½_{n}n(2t_{1}+d(n-1)) |
For this arithmetic series: 5+7+9+11+13+15+17; the sum of the first 7 terms is: S_{7}=½⋅7⋅(5+17)S_{7}=½⋅7⋅(22)S_{7}=77 |

Determine the general term, t, for a geometric sequence. (1.3, 1.4)_{n} |
A rule is:t=_{n}t_{1}⋅r^{n-1}where t_{1} is the first term, r is the common ratio, and n is the number of terms. |
For this geometric sequence: 1, -0.25, 0.0625, … the 6th term is: t_{6}=(-0.25)^{6-1}t_{6}=(-1)^{5}⋅(0.25)^{5}t_{6}=-0.000 976 5… |

Determine the sum of n terms, S, of a geometric series. (1.4)_{n} |
A rule is: where t_{1} is the first term, r is the common ratio, and n is the number of terms. |
For this geometric series: 4, 2, 1, … the sum of the first 10 terms is: or approximately 8 |

Determine the sum, S_{∞}, of a convergent infinite geometric series.(1.6) |
When r is between-1 and 1, use this rule:where t_{1} is the first term and r is the common ratio. |
For this geometric series: the sum is: |