Question

i. Write an A.P. in which a = 10 and d is any natural number.

ii. Find the sum of the first ten terms using formula.

iii. Can - 80 be a term of this A.P. ? Justify.​

Answer 1

Question :-

i) Write an A.P. in which a = 10 and d is any natural number.

ii) Find the sum of the first ten terms using formula.

iii) Can (-80) be a term of this A.P. ? Justify.

Solution :-

we know that,

• A sequence is said to be in AP (Arithmetic Progression), if the difference between its consecutive terms are equal.

• The nth term of an AP is given as ;

T(n) = a + (n-1)•d , where a is the first term and d is the common difference.

• The common difference of an AP is given as ;

d = T(n) - T(n-1)

• If the number of terms in an AP is n ( where n is odd ) ,then there will be a single middle term.

Also, [(n+1)/2]th term will be its middle term.

• If the number of terms in an AP is n ( where n is even ) ,then there will be two middle terms.

Also, (n/2)th and (n/2 + 1)th terms will be its middle terms.

• The sum up to nth terms of an AP is given as ;

S(n) = (n/2)•[2a + (n-1)•d] where a is the first term and d is the common difference.

• The nth term of an AP is also given as ;

T(n) = S(n) - S(n-1) .

given that,

• first term of AP = a = 10
• common difference = d (where d is a natural number.)

so,

→ Required AP will be = 10, (10+d) , (10 + 2d) , (10 + 3d) , ___________

and,

→ sum of first 10 terms of AP = (n/2)[2a + (n - 1)d] = (10/2)[2*10 + (10 - 1)d] = 5[20 + 9d] = (100 + 45d)

since, d is a natural number , therefore, we can conclude that, a negative number is not possible to be term of given AP .

therefore, (-80) is not a term of given AP.