# Sequence and Series

I. Arithmetic Sequence and Series

Arithmetic sequence is a sequence of numbers such that the difference between the consecutive terms is constant.

For example,  the sequence 2, 5, 8, 11, 14, 17, . . . is an arithmetic sequence with common difference of 3.

In an arithmetic sequence , First term is $$u_{1}$$ , Common difference is $$d$$ , General term of a sequence $$u_{n}$$, Sum of first n terms $$S_{n}$$

General form of an arithmetic sequence is $$u_{ 1 },\quad u_{ 1 }+d,\quad u_{ 1 }+2d,\quad u_{ 1 }+3d,\quad u_{ 1 }+4d,…………….u_{ 1 }+(n-1)d$$

The nth term or general term of arithmetic sequence is $${ u }_{ n }=u_{ 1 }+(n-1)d$$

Sum of the first terms of an arithmetic sequence is $$S_{ n }=\frac { n }{ 2 } [2u_{ 1 }+(n-1)×d]$$

If first term $$u_{1}$$and last term $$u_{n}$$ of an arithmetic series is given then the sum if the series is $$S_{ n }=\frac { n }{ 2 } [u_{ 1 }+u_{ n }]$$

That is Sum up to infinity in arithmetic sequence is infinite only, so sum up to infinity is not possible in arithmetic sequence.

II. Geometric Sequence and Series

Geometric sequence is a sequence of numbers such that the ratio or division of two consecutive terms is constant. For example,  the sequence 2, 6, 12, 24, 48, 96, . . . is a geometric sequence with common ratio of 3.

In a geometric sequence , First term is $$u_{1}$$ , Common ratio is $$r$$ , General term of a sequence $$u_{n}$$, Sum of first n terms $$S_{n}$$

General form of a geometric sequence is $$u_{ 1 },\quad u_{ 1 }×r,\quad u_{ 1 }×r^{ 2 },\quad u_{ 1 }×r^{ 3 },\quad u_{ 1 }×r^{ 4 },……………u_{ 1 }×{ r }^{ n-1 }$$

The nth term or general term of geometric sequence is $${ u }_{ n }\quad =\quad u_{ 1 }×{ r }^{ n-1 }$$

Sum of the first terms of a Geometric sequence is

$$S_{ n }=\frac { u_{ 1 }×(r^{ n }-1) }{ (r-1) }$$ if the common ratio | r | > 1

$$S_{ n }=\frac { u_{ 1 }\quad ×\quad (1-r^{ n }) }{ (1-r) }$$ if the common ratio | r | < 1

• Sum up to infinity in geometric sequence is possible only if the common ratio | r | < 1

$$S_{ \infty }=\frac { u_{ 1 } }{ (1-r) }$$

Skill Builder Questions

 Question no : 1The general term of an arithmetic sequence is . [a] Find the first 4 terms of the sequence.                                                                                      [b] Find the 50th term of the sequence. Question no : 2 The general term of an arithmetic sequence is . [a] Find the first term and common difference [b] Find the 20th term Question no : 3Consider the sequence 2, 6, 10, 14, . . . . . . . [a] Show that it’s an arithmetic sequence [b] Find the next three terms [c] Find the expression for the  term of the sequence [d] Hence find the 15th term of the sequence Question no : 4Consider the arithmetic sequence 20, 17, 14, 11, . . . . . . [a] Find the common difference [b] Find the 10th term of the sequence. [c]  Given that un=37 , find ‘n’ . Question no : 5Given that 16th term in an arithmetic sequence is 44 and the common difference is 3. [a] Find the first 4 terms of the sequence. [b] Find the 51st term of the sequence. [c] Find the sum of first 20 terms of the sequence. Question no : 6In an arithmetic sequence u10 = 43 and u33 = 204 find the first term and the common difference. Question no : 7Find the sum of the series 11 + 18 + 25 + 32 + …….. + 74 Question no : 8In an arithmetic sequence the first term is 3 and the 25th term is 51. Find the sum of the first 30 terms of the sequence. Question no : 9In an arithmetic sequence the first term is 16, nth term (un)  is 81 and the sum of the first terms (Sn)  is 679. Find the number of terms  in the sequence. Question no : 10In an arithmetic sequence 10th term is 109 and 38th term is 389. [a] Find the first term and the common difference. [b] Find the 20th term of the sequence. Question no : 11In an arithmetic sequence u14= -53  and u40 = -156 . Find the sum up to 50 terms of the series. Question no : 12In an arithmetic sequence u1 = -38 ,  un = -478  and  Sn = -11610  , find the number of terms n . Question no : 3Express the following sigma notation in expanded (series) form. Do not find the sum. Question no : 14Evaluate the sum of the following expressions. Question no : 15Find the missing terms in each of the following geometric sequence. [a]  _____ , 1 , _______ , 9                                  [b]  _____ , 4 , _______ , 36 [c]  3, _____ , _____ , _____ , 48                       [d]  2 , _____ ,_____ ,_____ ¸162 Question no : 16In a geometric sequence the first term is 5 and the common ratio is 2. [a] Find the 10th term. [b] Find the sum up to 7 terms. Question no : 17First three  terms of a sequence are 4 , 8 , 16 [a] show that it is a geometric sequence [b] Find the next three terms [c] Find the general term expression [d] hence find the 10th term of the sequence Question no : 18The first three terms of a geometric sequence are 250, 50, 10 . . . [a] Show that sum up to infinity exist  for this sequence [b] Find the next three terms [c] Find the sum up to infinity of the sequence. Question no : 19The first three terms of a series are -27 + 9 – 3 + . . . .  [a] Show that this is a infinite geometric series [b] Find the 7th term of the series [c] Find the sum of first 7 terms of the series [d] Find the sum of  the first 10 terms of the series [e] Find the sum up to infinity of the series. Question no : 20In a geometric sequence and  u1 = 3 and u4 = 375 [a] Find the common ration ‘ r ‘ . [b] Find the 8th term of the  sequence [c] Find S7