Arithmetic Progression (A.P.)

150 workers were engaged to finish a job in a certain number of days. 4 workers dropped out on second day, 4 more workers dropped out on third day and so on. It took 8 more days to finish the work. Find the number of days in which the work was completed.

Find the sum of odd integers from $1$ to $2001$.

A man deposited ₹$10000$  in a bank at the rate of $5\%$ simple interest annually. Find the amount in ${15}^{th}$ year since he deposited the amount and also calculate the total amount after  $20$ years.

Shamshad Ali buys a scooter for `₹ $22000$`. He pays `₹ $4000$` cash and agrees to pay the balance in an annual installment of `₹ $1000$` plus $10\%$ interest on the unpaid amount. How much will the scooter cost him?

A farmer buys a used tractor for Rs$12000$ . He pays Rs $6000$ cash and agrees to pay the balance in annual installments of Rs $500$ plus $12\%$ interest on the unpaid amount. How much will be the tractor cost him?

If $a,b,c$ are in A.P; $b,c,d$ are in G.P and $\frac{1}{c},\frac{1}{d},\frac{1}{e}$ are in A.P. prove that $a,c,e$ are in G.P.

The ratio of the A.M and G.M. of two positive numbers $a$ and $b$ is $m:n$. Show that $a:b=\left(m+\sqrt{m^2-n^2}\right):\left(m-\sqrt{m^2-n^2}\right)$

If  $a\left(\frac{1}{b}+\frac{1}{c}\right),b\left(\frac{1}{c}+\frac{1}{a}\right),c\left(\frac{1}{a}+\frac{1}{b}\right)$ are in A.P., prove that $a,b,c$ are in A.P.

The $p^{th}, q^{th}$ and $r^{th}$ terms of an A.P. are $a,b,c$ respectively. Show that $(q-r)a+\left(r-p\right)b+\left(p-q\right)c=0$

The sum of the first four terms of an $A.P.$ is $56$. The sum of the last four terms is $112$. If its first term is $11$, then find the number of terms.

Find the sum of all two-digit numbers which when divided by $4,$ yields $1$ as remainder.

Find the sum of integers from $1$ to $100$ that are divisible by $2$ or $5.$

Find the sum of all numbers between $200$ and $400$ which are divisible by $7$.

Let the sum of $n,2n,3n$ terms of an A.P. be $S_1,S_2$ and $S_3$ respectively, show that $S_3=3(S_2-S_1)$

If the sum of three numbers in $A.P.$ is $24$ and their product is $440$, find the numbers.

Show that the sum of ${\left(m+n\right)}^{th}$ and ${\left(m-n\right)}^{th}$ terms of an $A.P$ is equal to twice the $m^{th}$ term

The difference between any two consecutive interior angles of a polygon is $5°$. If the smallest angle is $120°$, find the number of the sides of the polygon.

A man starts repaying a loan as first installment of$₹100$. If he increases the installment by $₹5$ every month, what amount he will pay in the $30^{th}$ installment?

Between $1$ and $31$, $m$ numbers have been inserted in such a way that the resulting sequence is an A.P. and the ratio of $7^{th}$ and $(m-1{)}^{th}$ numbers is $5:9.$ Find the value of $m.$

If $\frac{a^n+b^n}{a^{n-1}+b^{n-1}}$ is the A.M. between $a$ and $b$, then find the value of $n.$