Answer :

In the given question we need to find the sum of the series.

For that, first, we need to find the n^{th} term of the series so that we can use summation of the series with standard identities and get the required sum.

The series given is 1 + 5 + 12 + 22 + 35 … to n terms.

This question can be solved by the method of difference.

__Note:__

Consider a sequence a_{1}, a_{2}, a_{3} …such that the Sequence a_{2} –a_{1}, a_{3} – a_{2}… is either an. A.P. or a G.P.

The n^{th} term, of this sequence, is obtained as follows:

S = a_{1} + a_{2} + a_{3} +…+ a_{n–1} + a_{n}→ (1)

S = a_{1} + a_{2} +…+ a_{n–2} + a_{n–1} + a_{n} → (2)

Subtracting (2) from (1),

We get, a_{n} = a_{1}+ [(a_{2}–a_{1}) + (a_{3}–a_{2}) +… (a_{n} – a_{n–1})].

Since the terms within the brackets are either in an A.P. or a G.P, we can find the value of a_{n} the n^{th} term.

Thus, we can find the sum of the n terms of the sequence as,

So,

By using the method of difference, we can find the n^{th} term of the expression.

S_{n} = 1 + 5 + 12 + 22 + 35 + ….. + a_{n}→ (1)

S_{n} = 1 + 5 + 12 + 22+ 35 + …. + a_{n}→ (2)

(1) – (2) → 0 = 1 + 4 + 7 + 10+ ….. - a_{n}

So, n^{th} term of the series,

a_{n} = 1 + 4 + 7 + 10 + ….

So, the n^{th} term form an AP, with the first term, a = 1; common difference, d = 3.

The required n^{th} term of the series is the same as the sum of n terms of AP.

Sum of n terms of an AP,

So, n^{th} term of the series,

Now, we need to find the sum of this series, S_{n.}

__Note:__

I. Sum of first n natural numbers, 1 + 2 +3+…n,

II. Sum of squares of first n natural numbers, 1^{2} + 2^{2} + 3^{2}+….n^{2},

III. Sum of cubes of first n natural numbers, 1^{3} + 2^{3} + 3^{3} +…..n^{3},

IV. Sum of a constant k, N times,

So, for the given series, we need to find,

From, the above identities,

So, Sum of the series,

Rate this question :

A person has 2 paRD Sharma - Mathematics

The sum of the seRD Sharma - Mathematics

How many terms ofRD Sharma - Mathematics

If <span lang="ENRD Sharma - Mathematics

Let a_{n}RD Sharma - Mathematics

Find the sum of 2RD Sharma - Mathematics

If <span lang="ENRS Aggarwal - Mathematics

Find the sum (41 RS Aggarwal - Mathematics

Find the sum (2 +RS Aggarwal - Mathematics

If<span lang="EN-RS Aggarwal - Mathematics