CBSE Class 10-Mathematics: Chapter –5 Arithmetic Progressions Part 18

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Question 3:

A sum of Rs is to be used to give seven cash prizes to students of a school for their overall academic performance. If, each prize is Rs less than its preceding term, find the value of each of the prizes.

Answer:

It is given that sum of seven cash prizes is equal to .

And each prize is R.s 20 less than its preceding term.

Let value of first prize Rs. a

Let value of second prize

Let value of third prize

So, we have sequence of the form:

It is an arithmetic progression because the difference between consecutive terms is constant.

First term , Common difference

Applying formula, to find sum of n terms of AP, we get

Therefore, value of first prize

Value of second prize

Value of third prize

Value of fourth prize

Value of fifth prize

Value of sixth prize

Value of seventh prize

Question 4:

A spiral is made up of successive semicircles, with centers alternatively at A and B, starting with center at A, of radii , , , , … What is the total length of such a spiral made up of thirteen consecutive semicircle?

Answer:

Length of semi-circle

Length of semicircle of radii

Therefore, we have sequence of the form:

terms .

To find total length of the spiral, we need to find sum of the sequence terms

Total length of spiral terms

Total length of spiral terms … (1)

Sequence terms is an arithmetic progression.

Let us find the sum of this sequence.

First term , Common difference and

Applying formula, to find sum of n terms of AP, we get

Therefore, terms

Putting this in equation (1), we get

Total length of spiral