# Arithmetic and Geometric Progressions

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With some simple but very powerful ideas, one can cut down on a lot of working when it comes to progressions. For example, anchoring a progression around its middle term can be very useful. Reinforce these ideas with the following questions.
1. ### Geometric Progressions

Second term of a GP is 1000 and the common ratio is $r = \frac{1}{n}$ where n is a natural number. Pn is the product of n terms of this GP. P6 > P5 and P6 > P7, what is the sum of all possible values of n?

1. 4
2. 9
3. 5
4. 13
• Correct AnswerChoice (B): 9 Correct answer
• Explanatory Answer
• Counting and Progressions
• Hard
2. ### GP: Common Ratio

Sum of first 12 terms of a GP is equal to the sum of the first 14 terms in the same GP. Sum of the first 17 terms is 92, what is the third term in the GP?

1. 92
2. -92
3. 46
4. 231
3. ### AP: Sum up to 'n' Terms

Sum of first 25 terms in AP is 525, sum of the next 25 terms is 725, what is the common difference?

1. ${8 \over 25}$
2. ${4 \over 25}$
3. ${6 \over 25}$
4. ${1 \over 25}$
4. ### AP: Common Difference

Let the nth term of AP be defined as tn, and sum up to 'n' terms be defined as Sn. If |t8| = |t16| and t3 is not equal to t7, what is S23?

1. 23(t16 - t8)
2. 0
3. 23t11
4. Cannot be determined
5. ### AP: Mean

a, b, c, d and e are 5 distinct numbers that from an arithmetic progression. They are not necessarily consecutive terms but form the first 5 terms of the AP. It is known that c is the arithmetic mean of a and b, and d is the arithmetic mean of b anc c. Which of the following statements are true?

i. Average of all 5 terms put together is c.
ii. Average of d and e is not greater than average of a and b.
iii. Average of b and c is greater than average of a and d.

1. i and ii only
2. ii and iii only
3. all three statements are true
4. i and iii only
6. ### GP Median

Consider a, b, c in a G.P. such that |a + b + c| = 15. The median of these three terms is a, and b = 10. If a > c, what is the product of the first 4 terms of this G.P.?

1. 40000
2. 32000
3. 8000
4. 48000
7. ### Arithmetic Progression

If 4 times the 4th term of an A.P. is equal to 9 times the 9th term of the A.P., what is 13 times the 13th term of this A.P.?

1. 7 times the 13th term
2. 0
3. 13 times the 7th term
4. 4 times the 4th term + 9 times the 9th term
8. ### Arithmetic Progression

Sequence P is defined by pn = pn-1 + 3, p1 = 11, Sequence Q is defined as qn = qn-1 – 4, q3 = 103. If pk > qk+2, what is the smallest value k can take?

1. 6
2. 11
3. 14
4. 15
9. ### Arithmetic Progression

The sum of 2n terms of A.P. {1, 5, 9, 13…..} is greater than sum of n terms of A.P. = {56, 58, 60..…}. What is the smallest value n can take?

1. 9
2. 10
3. 12
4. 14
10. ### Arithmetic Progression

a, b, c and d are in A.P., What can we say about terms bcd, acd, abd and abc?

1. They are also in A.P.
2. They are also in H.P.
3. They are also in G.P.
4. They are not in an A.P., G.P. or H.P.
• Correct AnswerChoice (B).
They are also in H.P.
Correct answer
• Explanatory Answer
• Arithmetic Progression
• Medium
11. ### Arithmetic Progression

Second term in an AP is 8 and the 8th term is 2 more than thrice the second term. Find the sum up to 8 terms of this AP.

1. 124
2. 108
3. 96
4. 110
• Correct AnswerChoice (A).
124
Correct answer
• Explanatory Answer
• Arithmetic Progression
• Medium
12. ### Sum of infinite terms

If ${S}_{n} = {n}^{3} + {n}^{2} + n + 1$ , where ${S}_{n}$ denotes the sum of the first n terms of a series and ${t}_{m}$ = 291, then m is equal to?

1. 24
2. 30
3. 26
4. 20
13. ### Sum of infinite terms

Sum of infinite terms of a GP is 12. If the first term is 8, what is the 4th term of this GP?

1. $\frac{8}{27}$

2. $\frac{4}{27}$

3. $\frac{8}{20}$

4. $\frac{1}{3}$

14. ### Sum of a Sequence

Find sum : $2^{2} + 2 * 3^{2} + 3 * 4^{2} + 4 * 5^{2}.....10 * 11^{2}$

1. 6530
2. 3600
3. 2850
4. 3850
15. ### Ratio of Amounts

The salaries earned by two friends Anil and Jeetu in different years are in A.P. If the ratio of the amount earned by them in ‘p’ number of years are (4p+1) : (2p+17). Then find the ratio of amount earned by them in the 7th year.

1. (2p+1) : (4p+6)
2. 53 : 43
3. 4 : 7
4. 15p : 36p
16. ### Investments

Ram invests a total sum of 2000 rupees on government bonds in 4 years. If these investments are in A.P and the sum of squares of the investments is 1200000. Find the investment made by ram in each year respectively. It is also known that he always invest more than the previous year.

1. 200,400,600,800
2. 875,625,375,125
3. 125,375,625,875
4. 50,350,650,950
• Correct AnswerChoice (A).
200,400,600,800
Correct answer
• Explanatory Answer
• Arithmetic Progressions
• Medium
17. ### Total Amount

Ram invest different amounts during the year on shares. S1, S2, S3……….Sm are different sums of ‘n’ amounts invested in ‘m’ years. If the amounts invested during the years are in A.P whose first terms are 1,2,3…..m and common difference are 1,3,5…..,(2m-1) respectively then find the total amount invested by Ram in ‘m’ years.

1. n(m+1)
2. m+1
3. $\frac{mn}{2}$(mn+1)
4. cannot be determined
18. ### Sum of Series

Find the sum of the series .4 + .44 + .444……. to n terms

1. 5.69
2. 14.44
3. $\frac{4}{81}[9n-1+\frac{1}{10^n}]$
4. $\frac{4}{81}[n + 1]$
19. ### Type of Progression

If the equation px2 + 2qx + r = 0 and dx2 + 2ex + f = 0 have a common root, then in which type of progression is $\frac{d}{p} , \frac{e}{q} , \frac{f}{r}$

1. $\frac{d}{p} , \frac{e}{q} , \frac{f}{r}$ are in G.P
2. $\frac{d}{p} , \frac{e}{q} , \frac{f}{r}$ are in A.P
3. $\frac{d}{p} , \frac{e}{q} , \frac{f}{r}$ are in H.P
4. Insufficient Data
20. ### Sum of all Terms

Find the sum of all the terms, If the first 3 terms among 4 positive integers are in A.P and the last 3 terms are in G.P. Moreover the difference between the first and last term is 40.

1. 108
2. 172
3. 124
4. 196

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