Arithmetic Progressions

Study Notes

An arithmetic progression (AP) is a sequence of numbers in which each term is obtained by adding a fixed number to the previous term.

Exercises on Arithmetic Progressions

Find the sum of the following APs:
2, 7, 12,..., to 10 terms.
-37, -33, -29,..., to 12 terms.
1.5, 1.2, 1.0,..., to 11 terms.
0.6, 1.7, 2.8,..., to 100 terms.
Find the sums of the given series:
7 + 10 + 14 +...+ 84.
34 + 32 + 30 +...+ 10.

Problems Involving APs

In an AP:
given a = 5, d = 3, an = 50, find n and Sn.
given a = 7, a13 = 35, find d and S13.
given a12 = 37, d = 3, find a and S12.
given a1 = 15, S10 = 125, find d and a10.
given d = 5, S9 = 75, find a and a9.
given a = 2, d = 8, Sn = 90, find n and an.
given a = 8, an = 62, Sn = 210, find n and d.
given a = 4, d = 2, Sn = -14, find n and a.
given a = 3, n = 8, Sn = 192, find d.
given l = 28, Sn = 144, and there are total 9 terms, find a.

Applications of APs

Find the number of terms of the AP: 9, 17, 25,..., to give a sum of 636.
The first term of an AP is 5, the last term is 45, and the sum is 400. Find the number of terms and the common difference.
The first and last terms are 17 and 350, respectively, and the common difference is 9. Find the number of terms and their sum.
Find the sum of the first 22 terms of an AP with d = 7 and 22nd term = 149.
Find the sum of the first 51 terms of an AP with second and third terms = 14 and 18, respectively.
If the sum of the first 7 terms of an AP is 49 and that of 17 terms is 289, find the sum of the first n terms.
Show that a,..., a form an AP where ai is defined as:
a = 3 + 4n.
a = 9 - 5n.
Find the sum of the first 15 terms in each case.
If the sum of the first n terms of an AP is 4n - n², find the first term, the sum of the first two terms, the second term, and the nth term.
Find the sum of the first 40 positive integers divisible by 6.
Find the sum of the first 15 multiples of 8.
Find the sum of the odd numbers between 0 and 50.
A contractor has to pay a penalty of ₹200 for the first day, ₹250 for the second day, ₹300 for the third day, and so on, with a penalty of ₹50 more for each succeeding day. Find the penalty if the contractor has delayed the work by 30 days.
A sum of ₹700 is to be used to give seven cash prizes to students. If each prize is ₹20 less than its preceding prize, find the value of each prize.
A school has decided to plant trees in and around the school. The number of trees to be planted by each section of each class is equal to the class number. If there are three sections of each class, find the number of trees to be planted.
A spiral is made up of successive semicircles with radii 0.5 cm, 1.0 cm, 1.5 cm, 2.0 cm, ... . Find the total length of the spiral made up of 13 consecutive semicircles.

Logs and Potatoes

200 logs are stacked in the following manner: 20 logs in the bottom row, 19 in the next row, and so on. Find the number of rows and the number of logs in the top row.
In a potato race, a bucket is placed at the starting point, and the potatoes are placed 3 m apart. Find the total distance the competitor has to run to pick up all the potatoes.

Elimination Method for Solving Equations

The elimination method is used to solve a system of linear equations.
Steps to solve the equations:
Multiply the equations by suitable multiples to make the coefficients of one variable equal.
Subtract one equation from the other to eliminate the variable.
Solve the resulting equation to find the value of the other variable.
Substitute the value of the variable into one of the original equations to find the value of the other variable.
Verify the solution by checking if it satisfies both equations.

Description

Solve various problems involving arithmetic progressions, including finding sums, identifying terms, and determining properties. Exercises cover different types of APs and require applying formulas and concepts.