First Principles AI

Question 1

Sum the series:

(i)

$3 + 6 + 9 + \cdots + a_{20}$

(ii)

$\dfrac{4}{\sqrt{5}} + \sqrt{5} + \dfrac{6}{\sqrt{5}} + \cdots + a_n$

Solution Theory

Question 2

Find $S_n$ for each arithmetic series:

(i)

$a_1 = 4,\ n = 25,\ a_n = 100$

(ii)

$a_1 = 40,\ n = 20,\ d = -3$

(iii)

$a_n = 52,\ n = 21,\ d = -4$

Solution Theory

Question 3

Find $a_1$ for the arithmetic series: $d = 8,\ n = 19,\ S_n = 1786$ .

Solution Theory

Question 4

How many terms of the series: $96 + 93 + 90 + \cdots$ amount to 1071.

Solution Theory

Question 5

If the three sides of a right-angled triangle having perimeter 36 cm are in A.P., find them.

Solution Theory

Question 6

Sum the series

(i)

$3 + 5 - 7 + 9 + 11 - 13 + 15 + 17 - 19 + \cdots$ to $3n$ terms.

(ii)

$1 + 4 - 7 + 10 + 13 - 16 + 19 + 22 - 25 + \cdots$ to $3n$ terms.

Solution Theory

Question 7

Find the sum of 20 terms of the series whose $r^{th}$ term is $3r + 1$ .

Solution Theory

Question 8

The 5th and 9th terms of an A.P. are 11 and 17 respectively. Find the sum of 20 terms.

Solution Theory

Question 9

Obtain the sum of all integers in the first 1000 positive integers which are neither divisible by 5 nor by 2.

Solution Theory

Question 10

The sum of 9 terms of an A.P. is 171 and its eighth term is 31. Find the series.

Solution Theory

Question 11

The 5th term of an arithmetic progression is 21 and the sum of first six terms is 90. Find the 18th term.

Solution Theory

Question 12

The sum of three numbers in an A.P. is 24 and their product is 440. Find the numbers.

Solution Theory

Question 13

The first four terms of an A.P. are 2, 6, 10 and 14. Find the least number of terms needed so that the sum of the terms is greater than 2000.

Solution Theory

Question 14

Find four numbers in A.P. whose sum is 32 and the sum of whose squares is 276.

Solution Theory

Question 15

Find the five numbers in A.P. whose sum is 25 and the sum of whose squares is 135.

Solution Theory

Question 16

If $\dfrac{1}{a+b},\ \dfrac{1}{c+a},\ \dfrac{1}{b+c}$ are in A.P. then show that $a^2,\ b^2,\ c^2$ are in A.P.

Solution Theory

Question 17

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 number of terms.

Solution Theory

Question 18

The first, second and last terms of an A.P. are $a,\ b$ and $c$ respectively. Show that the sum of A.P. is $\dfrac{(b+c-2a)(c+a)}{2(b-a)}$ .

Solution Theory

Question 19

Show that the sum of $n$ A.Ms. between $a$ and $b$ is $n$ times the single A.M. between them.

Solution Theory

Questions