**Algebraic Expressions**

**1. Observe the patterns of digits made from line segments of equal length. You will find such segmented digits on the display of electronic watches or calculators.**

**If the number of digits formed is taken to be n, the number of segments required to form n digits is given by the algebraic expression appearing on the right of each pattern. How many segments are required to form 5, 10, 100 digits of the kind **

**Solution:-**

**(a) From the question it is given that the numbers of segments required to form n digits of the kind**is (5n + 1)

Then,

The number of segments required to form 5 digits = ((5 × 5) + 1)

= (25 + 1)

= 26

The number of segments required to form 10 digits = ((5 × 10) + 1)

= (50 + 1)

= 51

The number of segments required to form 100 digits = ((5 × 100) + 1)

= (500 + 1)

= 501

**(b) From the question it is given that the numbers of segments required to form n digits of the kind**is (3n + 1)

Then,

The number of segments required to form 5 digits = ((3 × 5) + 1)

= (15 + 1)

= 16

The number of segments required to form 10 digits = ((3 × 10) + 1)

= (30 + 1)

= 31

The number of segments required to form 100 digits = ((3 × 100) + 1)

= (300 + 1)

= 301

**(c) From the question it is given that the numbers of segments required to form n digits of the kind**is (5n + 2)

Then,

The number of segments required to form 5 digits = ((5 × 5) + 2)

= (25 + 2)

= 27

The number of segments required to form 10 digits = ((5 × 10) + 2)

= (50 + 2)

= 52

The number of segments required to form 100 digits = ((5 × 100) + 1)

= (500 + 2)

= 502

**2. Use the given algebraic expression to complete the table of number patterns.**

S. No. | Expression | Terms | |||||||||

1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | … | 10^{th} | … | 100^{th} | … | ||

(i) | 2n – 1 | 1 | 3 | 5 | 7 | 9 | – | 19 | – | – | – |

(ii) | 3n + 2 | 5 | 8 | 11 | 14 | – | – | – | – | – | – |

(iii) | 4n + 1 | 5 | 9 | 13 | 17 | – | – | – | – | – | – |

(iv) | 7n + 20 | 27 | 34 | 41 | 48 | – | – | – | – | – | – |

(v) | n^{2} + 1 | 2 | 5 | 10 | 17 | – | – | – | – | 10001 | – |

**Solution:-**

**(i) From the table (2n – 1)**

Then, 100^{th }term =?

Where n = 100

= (2 × 100) – 1

= 200 – 1

= 199

**(ii) From the table (3n + 2)**

5^{th }term =?

Where n = 5

= (3 × 5) + 2

= 15 + 2

= 17

Then, 10^{th }term =?

Where n = 10

= (3 × 10) + 2

= 30 + 2

= 32

Then, 100^{th }term =?

Where n = 100

= (3 × 100) + 2

= 300 + 2

= 302

**(iii) From the table (4n + 1)**

5^{th }term =?

Where n = 5

= (4 × 5) + 1

= 20 + 1

= 21

Then, 10^{th }term =?

Where n = 10

= (4 × 10) + 1

= 40 + 1

= 41

Then, 100^{th }term =?

Where n = 100

= (4 × 100) + 1

= 400 + 1

= 401

**(iv) From the table (7n + 20)**

5^{th }term =?

Where n = 5

= (7 × 5) + 20

= 35 + 20

= 55

Then, 10^{th }term =?

Where n = 10

= (7 × 10) + 20

= 70 + 20

= 90

Then, 100^{th }term =?

Where n = 100

= (7 × 100) + 20

= 700 + 20

= 720

**(v) From the table (n ^{2} + 1)**

5^{th }term =?

Where n = 5

= (5^{2}) + 1

= 25+ 1

= 26

Then, 10^{th }term =?

Where n = 10

= (10^{2}) + 1

= 100 + 1

= 101

So the table is completed below.

S. No. | Expression | Terms | |||||||||

1^{st} | 2^{nd} | 3^{rd} | 4^{th} | 5^{th} | … | 10^{th} | … | 100^{th} | … | ||

(i) | 2n – 1 | 1 | 3 | 5 | 7 | 9 | – | 19 | – | 199 | – |

(ii) | 3n + 2 | 5 | 8 | 11 | 14 | 17 | – | 32 | – | 302 | – |

(iii) | 4n + 1 | 5 | 9 | 13 | 17 | 21 | – | 41 | – | 401 | – |

(iv) | 7n + 20 | 27 | 34 | 41 | 48 | 55 | – | 90 | – | 720 | – |

(v) | n^{2} + 1 | 2 | 5 | 10 | 17 | 26 | – | 101 | – | 10001 | – |