NCERT Solutions · Class 10 Mathematics Arithmetic Progressions

If the sum of the first $\displaystyle n$ terms of an AP is $\displaystyle 4 n-n^{2}$, what is the first term (that is $\displaystyle \mathrm{S}_{1}$ )? What is the sum of first two terms? What is the second term? Similarly, find the 3rd, the 10th and the $\displaystyle n$th terms.

Find the sum of the first $\displaystyle 40$ positive integers divisible by 6.

Find the sum of the first $\displaystyle 15$ multiples of 8.

Find the sum of the odd numbers between $\displaystyle 0$ and 50.

A contract on construction job specifies a penalty for delay of completion beyond a certain date as follows: ₹ $\displaystyle 200$ for the first day, ₹ $\displaystyle 250$ for the second day, ₹ $\displaystyle 300$ for the third day, etc., the penalty for each succeeding day being ₹ $\displaystyle 50$ more than for the preceding day. How much money the contractor has to pay as penalty, if he has delayed the work by $\displaystyle 30$ days?

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

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 $\displaystyle 1$ tree, a section of Class II will plant $\displaystyle 2$ trees and so on till Class XII. There are three sections of each class. How many trees will be planted by the students?

A spiral is made up of successive semicircles, with centres alternately at A and B, starting with centre at A, of radii $\displaystyle 0.5$ cm, $\displaystyle 1.0$ cm, $\displaystyle 1.5$ cm, $\displaystyle 2.0$ cm, ... as shown in Fig. 5.4. What is the total length of such a spiral made up of thirteen consecutive semicircles? $\displaystyle \left(\right.$ Take $\displaystyle \left.\pi=\frac{22}{7}\right)$ [Hint : Length of successive semicircles is $\displaystyle l_{1}, l_{2}, l_{3}, l_{4}, \ldots$ with centres at $\displaystyle \mathrm{A}, \mathrm{B}, \mathrm{A}, \mathrm{B}, \ldots$, respectively.]

$\displaystyle 200$ logs are stacked in the following manner: $\displaystyle 20$ logs in the bottom row, $\displaystyle 19$ in the next row, $\displaystyle 18$ in the row next to it and so on (see Fig. $\displaystyle 5.5$). In how many rows are the $\displaystyle 200$ logs placed and how many logs are in the top row?

In a potato race, a bucket is placed at the starting point, which is $\displaystyle 5$ m from the first potato, and the other potatoes are placed $\displaystyle 3$ m apart in a straight line. There are ten potatoes in the line (see Fig. $\displaystyle 5.6$). 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? [Hint : To pick up the first potato and the second potato, the total distance (in metres) run by a competitor is $\displaystyle 2 \times 5+2 \times(5+3)$ ]