Answer

Verified

411.9k+ views

**Hint:**To solve this question, what we will find the sum of the first 11 terms of A.P and then put 110 equal to the formula of the sum of A.P and then from there we will get the relation between a and d. then we will find the ${{6}^{th}}$ term of A.P and then we will reciprocal the value which will be equal to ${{6}^{th}}$ term of H.P.

**Complete step-by-step solution:**Before we solve this question, let’s see what are Arithmetic Progression and Harmonic Progression.

Arithmetic Progression is a series whose consecutive elements have a constant difference.

Denoted as a, a+d, a+2d, a+3d, ….., a+(n-1)d, where a = first term, d = difference between consecutive terms and n = number of terms in A.P

Harmonic progression is a series containing, reciprocal of terms of an A.P

Denoted as $\dfrac{1}{a}'\dfrac{1}{a+d}'\dfrac{1}{a+3d}'......'\dfrac{1}{a+(n-1)d}$ , $\dfrac{1}{a}$ = first term, n = number of terms in A.P

Now, it is given that the reciprocal of the first 11 terms of H.P terms sums 110. And obviously, the reciprocal of H.P will give us A.P.

So, sum of first 11 terms of an A.P = 110

Sum of first n terms $=\dfrac{n}{2}\left( 2a+(n-1)d \right)$

$110=\dfrac{n}{2}\left( 2a+(n-1)d \right)$

As, n = 11

So, $110=\dfrac{11}{2}\left( 2a+10d \right)$

On solving, we get

$10=\left( a+5d \right)$…….)( i)

Now, ${{n}^{th}}$ term of an A.P = a + (n-1)d

So, ${{6}^{th}}$ term of A.P = a + (6-1)d = a + 5d

From, ( i ), ${{6}^{th}}$ term of A.P = 10

As, terms of H.P is reciprocal of terms of A.P,

**So, ${{6}^{th}}$ term of H.P = $\dfrac{1}{10}$.**

**Note:**As we see on the question is asked in very clever language so one must know the concept of A.P and H.P series deeply. All formulas related to A.P and H.P must be known to solve the questions. Do not forget to convert terms of A.P in terms of H.P by doing reciprocal of terms of A.P.

Recently Updated Pages

The base of a right prism is a pentagon whose sides class 10 maths CBSE

A die is thrown Find the probability that the number class 10 maths CBSE

A mans age is six times the age of his son In six years class 10 maths CBSE

A started a business with Rs 21000 and is joined afterwards class 10 maths CBSE

Aasifbhai bought a refrigerator at Rs 10000 After some class 10 maths CBSE

Give a brief history of the mathematician Pythagoras class 10 maths CBSE

Trending doubts

Difference Between Plant Cell and Animal Cell

Give 10 examples for herbs , shrubs , climbers , creepers

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Name 10 Living and Non living things class 9 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Write a letter to the principal requesting him to grant class 10 english CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE