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**1. A.G.P. Properties Formula**

**Arithmetic Geometric Series**

A series will be an arithmetic-geometric series if each of its terms is formed by the product of the corresponding terms of an A.P and G.P.

The general form of A.G.P series is a, (a+d)r, (a+2d)$$r^{2}$$,......

Sum of ‘n’ terms of A.G.P series

$$S_{n}$$=$$\frac{a}{1-r}$$+rd$$\frac{(1-r^{n-1})}{1-r}$$+rn$$\frac{[a+(n-1)d]}{1-r}$$(r≠1)

Sum of infinite terms of A.G.P series

$$S_{∞}$$=$$\frac{a}{1-r}+\frac{dr}{(1-r)^{2}}$$(|r|<1)

**2. G.P. - Formulas and Properties**

**Geometric Progression**

If in a succession of numbers the ratio of any term and the previous term is constant then that numbers are said to be in Geometric Progression.

Ex :1, 3, 9, 27 or a, ar, a$$r^{2}$$, a$$r^{3}$$

The general expression of a G.P, Tn = a $$r^{n-1}$$ (where a is the first term and ‘r’ is the common ratio).

Sum of ‘n’ terms in G.P, Sn = $$\frac{a(1-r^{n})}{1-r}$$ (if r<1) or $$\frac {a(r^{n}-1)}{r-1}$$ (if r>1)

**Properties of G.P**

If a, b , c, d,.... are in G.P and ‘k’ is a constant then

- ak, bk, ck,...will also be in G.P
- a/k, b/k, c/k will also be in G.P

Sum of term of infinite series in G.P, $$S_{∞}$$=$$\frac {a}{1-r}$$ (-1 < r <1)

3. A.P. - Formulas and Properties

**Arithmetic progression**** (A.P)**

If the sum of the difference between any two consecutive terms is constant then the terms are said to be in A.P

Example: 2,5,8,11 or a, a+d, a+2d, a+3d...

If 'a' is the first term and 'd' is a common difference then the general 'n' term is $$T_{n}$$=a+(n-1)d

Sum of first 'n' terms in A.P=$$\frac{n}{2}$$[2a+(n-1)d]

Number of terms in A.P=$$\frac{Last Term-First Term}{Common Difference}$$+1

**Properties of Arithmetic progression**

If a, b, c, d,.... are in A.P and ‘k’ is a constant then

a-k, b-k, c-k,... will also be in A.P

ak, bk, ck,...will also be in A.P

a/k, b/k, c/k will also be in A.P

4. Harmonic Mean Formula

**Harmonic Mean**

If a, b, c, d...are the given numbers in H.P then the Harmonic mean of 'n' terms=$$\frac{Number of terms}{\frac{1}{a}+\frac{1}{a}+\frac{1}{a}+....}$$

If two numbers a and b are in H.P then the Harmonic mean= $$\frac{2ab}{a+b}$$

Let both the series $$a_{1},a_{2},a_{3}$$... and $$b_{1},b_{2},b_{3}$$... be in arithmetic progression such that the common differences of both the series are prime numbers. If $$a_{5}=b_{9},a_{19}=b_{19}$$ and $$b_{2}=0$$, then $$a_{11}$$ equals

correct answer:-**2**

The value of $$1 + \left(1 + \frac{1}{3}\right)\frac{1}{4} + \left(1 + \frac{1}{3} + \frac{1}{9}\right)\frac{1}{16} + \left(1 + \frac{1}{3} + \frac{1}{9} + \frac{1}{27}\right)\frac{1}{64} + -------$$ is

correct answer:-**4**

Let $$a_n = 46 + 8n$$ and $$b_n = 98 + 4n$$ be two sequences for natural numbers $$n \leq 100$$. Then, the sum of all terms common to both the sequences is

correct answer:-**1**

A lab experiment measures the number of organisms at 8 am every day. Starting with 2 organisms on the first day, the number of organisms on any day is equal to 3 more than twice the number on the previous day. If the number of organisms on the nth day exceeds one million, then the lowest possible value of n is

correct answer:-**19**

Let $$a_{n}$$ and $$b_{n}$$ be two sequences such that $$a_{n}=13+6(n-1)$$ and $$b_{n}=15+7(n-1)$$ for all natural numbers n. Then, the largest three digit integer that is common to both these sequences, is

correct answer:-**967**

The average of a non-decreasing sequence of N numbers $$a_{1},a_{2}, ... , a_{N}$$ is 300. If $$a_1$$, is replaced by $$6a_{1}$$ , the new average becomes 400. Then, the number of possible values of $$a_{1 }$$, is

correct answer:-**14**

For any natural number n, suppose the sum of the first n terms of an arithmetic progression is $$(n + 2n^2)$$. If the $$n^{th}$$ term of the progression is divisible by 9, then the smallest possible value of n is

correct answer:-**3**

On day one, there are 100 particles in a laboratory experiment. On day n, where $$n\ge2$$, one out of every n articles produces another particle. If the total number of particles in the laboratory experiment increases to 1000 on day m, then m equals

correct answer:-**1**

Consider the arithmetic progression 3, 7, 11, ... and let $$A_n$$ denote the sum of the first n terms of this progression. Then the value of $$\frac{1}{25} \sum_{n=1}^{25} A_{n}$$ is

correct answer:-**1**

Consider a sequence of real numbers, $$x_{1},x_{2},x_{3},...$$ such that $$x_{n+1}=x_{n}+n-1$$ for all $$n\geq1$$. If $$x_{1}=-1$$ then $$x_{100}$$ is equal to

correct answer:-**4**

For a sequence of real numbers $$x_{1},x_{2},...x_{n}$$, If $$x_{1}-x_{2}+x_{3}-....+(-1)^{n+1}x_{n}=n^{2}+2n$$ for all natural numbers n, then the sum $$x_{49}+x_{50}$$ equals

correct answer:-**4**

If $$x_0 = 1, x_1 = 2$$, and $$x_{n + 2} = \frac{1 + x_{n + 1}}{x_n}, n = 0, 1, 2, 3, ......,$$ then $$x_{2021}$$ is equal to

correct answer:-**4**

The natural numbers are divided into groups as (1), (2, 3, 4), (5, 6, 7, 8, 9), ….. and so on. Then, the sum of the numbers in the 15th group is equal to

correct answer:-**1**

If $$x_1=-1$$ and $$x_m=x_{m+1}+(m+1)$$ for every positive integer m, then $$X_{100}$$ equals

correct answer:-**1**

Let $$a_1, a_2, ...$$ be integers such that

$$a_1 - a_2 + a_3 - a_4 + .... + (-1)^{n - 1} a_n = n,$$ for all $$n \geq 1.$$

Then $$a_{51} + a_{52} + .... + a_{1023}$$ equals

correct answer:-**2**

If the population of a town is p in the beginning of any year then it becomes 3 + 2p in the beginning of the next year. If the population in the beginning of 2019 is 1000, then the population in the beginning of 2034 will be

correct answer:-**4**

If $$a_1 + a_2 + a_3 + .... + a_n = 3(2^{n + 1} - 2)$$, for every $$n \geq 1$$, then $$a_{11}$$ equals

correct answer:-**6144**

If $$(2n + 1) + (2n + 3) + (2n + 5) + ... + (2n + 47) = 5280$$, then whatis the value of $$1 + 2 + 3 + .. + n?$$

correct answer:-**4851**

The value of the sum 7 x 11 + 11 x 15 + 15 x 19 + ...+ 95 x 99 is

correct answer:-**80707**

If $$a_{1}=\frac{1}{2\times5},a_{2}=\frac{1}{5\times8},a_{3}=\frac{1}{8\times11},...,$$ then $$a_{1}+a_{2}+a_{3}+...+a_{100}$$ is

correct answer:-**1**

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