Vaccines produces antibody in our body to fight against disease causing microbes entering our body. Diseases like cholera, tuberculosis, smallpox and hepatitis can be prevented by vaccination; polio drop given to children is also a vaccine.

Antibiotics are medicines that kill or stop the growth of disease causing microorganisms and they are prepared from microorganisms. For e.g. streptomycin, tetracycline and penicillin are some of the antibiotics prepared from microorganisms like fungi and bacteria.

Fermentation is the process of food processing in which sugar is converted into alcohol by the action of microorganisms. This process is used to produce alcoholic beverages such as wine, beer, and cider. For this purpose yeast is grown on natural sugars present in grains like barley, wheat, rice, crushed fruit juices etc.

Yeast reproduces rapidly and produces carbon dioxide during respiration. Bubbles of gas fill the dough (mixture of atta or maida and some sugar and water) and increase its volume. This is the basis of using yeast in baking industry.

Bacterium lactobacillus multiplies in milk and converts it into curd.

Microorganisms are beneficial to us in various ways; they not only prepare curd, bread, cake, wine and medicines for us but also used to increase soil fertility by fixing atmospheric nitrogen.
Thus we called them as friendly.

Protozoa:

Protozoa had been defined as unicellularprotists with animal-like behaviour like movement. The largest protozoa known are the deep-sea dwellingxenophyophores, which can grow up to 20 cm in diameter.They move around with whip-like tails called flagella, and hair-like structures called cilia,it is responsible for causing diseases like dysentery and malaria in human beings.

Algae:

Algae are a very large and diverse group of simple, typicallyautotrophic organisms, ranging from unicellular to multicellular forms. Most are photosynthetic like plants, and "simple" because their tissues are not organized into many distinct organs found in land plants. The largest and most complex marine forms are called seaweeds.Algae exhibit a wide range of reproductive strategies, from simple, asexual cell division to complex forms of sexual reproduction.Nearly all algae have photosynthetic machinery ultimately derived from cyanobacteria, and so produce oxygen as a by-product of photosynthesis.

Bacteria:

Bacteria are single- celled microscopic organisms. They can survive under all types of environment, ranging from ice cold climate to hot springs and deserts to marshy lands. Bacteria always live in colonies. They are of spiral shape or rod shape. Bacteria play important role in our life; some bacteria are useful whereas some other are harmful and cause diseases. Bacteria are involved in making of cheese and pickles. Lactobacillus bacteria promote the formation of curd. Antibiotics are also made from bacteria. Apart from this diseases like tuberculosis and typhoid are caused due to bacteria.

Viruses:

Viruses are microscopic infectious agent that acts as non-living outside host cell and inside host cell becomes living and show reproduction. It can affect all kind of organism including animals, plants and bacteria. Common ailments like cold, coughs and influenza (flu) are caused by viruses; serious diseases like chicken pox and polio are also caused by viruses.

Here, a = 21, d = 18 – 21 = – 3 and an = – 81, and we have to find n.

As an = a + ( n – 1) d, we have – 81 = 21 + (n – 1)(– 3) – 81 = 24 – 3n – 105 = – 3n

So, n = 35 Therefore, the 35th term of the given AP is – 81.

Next, we want to know if there is any n for which an = 0.

If such an n is there, then 21 + (n – 1) (–3) = 0, i.e., 3(n – 1) = 21 i.e., n = 8

So, the eighth term is 0.  

Let us consider a positive integer a

Divide the positive integer a by 3, and let r be the reminder and b be the quotient such that

a = 3b + r……………………………(1)

where r = 0,1,2,3…..

Case 1: Consider r = 0

Equation (1) becomes

a = 3b

On squaring both the side

a2 = (3b)2

a2 = 9b2

a2 = 3 × 3b2

a2 = 3m

Where m = 3b2

Case 2: Let r = 1

Equation (1) becomes

a = 3b + 1

Squaring on both the side we get

a2 = (3b + 1)2

a2 = (3b)2 + 1 + 2 × (3b) × 1

a2 = 9b2 + 6b + 1

a2 = 3(3b2 + 2b) + 1

a2 = 3m + 1

Where m = 3b2 + 2b

Case 3: Let r = 2

Equation (1) becomes

a = 3b + 2

Squaring on both the sides we get

a2 = (3b + 2)2

a2 = 9b2 + 4 + (2 × 3b × 2)

a2 = 9b2 + 12b + 3 + 1

a2 = 3(3b2 + 4b + 1) + 1

a2 = 3m + 1

where m = 3b2 + 4b + 1

square of any positive integer is of the form 3m or 3m+1.

Hence proved.