The sum of the third and the seventh terms of an A P is 6 and their product is 8 . Find the sum of first sixteen terms of the A P .

We know that, a n = a + n - 1 d a 3 = a + 3 - 1 d ⇒ a 3 = a + 2 d Similarly, a 7 = a + 6 d Given that, a 3 + a 7 = 6 ⇒ a + 2 d + a + 6 d = 6 ⇒ 2 a + 8 d = 6 ⇒ a + 4 d = 3 ⇒ a = 3 - 4 d … i Also, it is given that a 3 × a 7 = 8 ⇒ a + 2 d × a + 6 d = 8 From equation ( i ) , 3 - 4 d + 2 d × 3 - 4 d + 6 d = 8 ⇒ 3 - 2 d × 3 + 2 d = 8 ⇒ 9 - 4 d 2 = 8 ⇒ 4 d 2 = 9 - 8 = 1 ⇒ d 2 = 1 4 ⇒ d = ± 1 2 ⇒ d = 1 2 or - 1 2 From equation i , ( When d is 1 2 ) a = 3 - 4 d ⇒ a = 3 - 4 1 2 = 3 - 2 = 1 ( When d is - 1 2 ) a = 3 - 4 - 1 2 ⇒ a = 3 + 2 = 5 S n = n 2 2 a + n - 1 d (When a is 1 and d is 1 2 ) S 16 = 16 2 2 1 + 16 - 1 1 2 = 8 2 + 15 2 = 4 19 = 76 ( When a is 5 and d is - 1 2 ) S 16 = 16 2 2 5 + 16 - 1 - 1 2 = 8 10 + 15 - 1 2 = 8 5 2 = 20

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A sum of $₹ 700$ is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is $₹ 20$ less than its preceding prize, find the value of each of the prizes.

Important Questions on Arithmetic Progressions

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.3>Q 17

In a school, students thought of planting trees in and around the school to reduce air pollution. It was decided that the number of trees, that each section of each class will plant, will be the same as the class, in which they are studying, e.g., a section of Class

I

will plant

1

tree, a section of Class

I I

will plant

2

trees and so on till Class

X I I

. There are three sections of each class. How many trees will be planted by the students ?

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.3>Q 18

A spiral is made up of successive semicircles, with centres alternately at $A$ and $B$ , starting with centre at $A$ , of radii $0.5 c m, 1.0 c m, 1.5 c m, 2.0 c m, . . .$ as shown in figure. What is the total length of such a spiral made up of thirteen consecutive semicircles? (Take $π = \frac{22}{7}$ )

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.3>Q 19

$200$ logs are stacked in the following manner: $20$ logs in the bottom row, $19$ in the next row, $18$ in the row next to it and so on (see Figure). In how many rows are the $200$ logs placed and how many logs are in the top row?

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.3>Q 20

In a potato race, a bucket is placed at the starting point, which is

5 m

from the first potato, and the other potatoes are placed

3 m

apart in a straight line. There are ten potatoes in the line (see Figure).
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A competitor starts from the bucket, picks up the nearest potato, runs back with it, drops it in the bucket, runs back to pick up the next potato, runs to the bucket to drop it in, and she continues in the same way until all the potatoes are in the bucket. What is the total distance the competitor has to run?

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.4>Q 1

Which term of the

A P

121, 117, 113, . . .,

is its first negative term?

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.4>Q 2

The sum of the third and the seventh terms of an

A P

6

and their product is

8

. Find the sum of first sixteen terms of the

A P

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.4>Q 3

A ladder has rungs

25 cm

apart. (See Fig.). The rungs decrease uniformly in length from

45 c m

at the bottom to

25 c m

at the top. If the top and the bottom rungs are

2 \frac{1}{2} m

apart, what is the length of the wood required for the rungs?

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MATHEMATICS Textbook for Class X>Chapter 5 - Arithmetic Progressions>Exercise 5.4>Q 4

The houses of a row are numbered consecutively from

1

49

. Show that there is a value of

x

such that the sum of the numbers of the houses preceding the house numbered

x

is equal to the sum of the numbers of the houses following it. Find this value of

x

A sum of ₹700 is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is ₹20 less than its preceding prize, find the value of each of the prizes.

Important Questions on Arithmetic Progressions

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A sum of $₹ 700$ is to be used to give seven cash prizes to students of a school for their overall academic performance. If each prize is $₹ 20$ less than its preceding prize, find the value of each of the prizes.