Arithmetic Progressions Practice Quiz

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Questions and Answers

What is the common difference of the AP: 3, 8, 13, 18, ...?

4 (correct)

Which term in the AP: 7, 13, 19, ..., 205 is calculated by the formula $a_n = a + (n-1)d$?

30th term

36th term

32nd term (correct)

34th term

What is the 31st term of an AP if the 11th term is 38 and 16th term is 73?

88 (correct)

If the 3rd term of an AP is 4 and the 9th term is -8, what is the common difference?

-3 Signup and view all the answers

Which term of the AP: 3, 15, 27, 39,... will be equal to 132 more than its 54th term?

90 Signup and view all the answers

How many terms are there in the AP: 18, 15, 13,...,-47?

26 Signup and view all the answers

If two APs have the same common difference and the difference between their 100th terms is 100, what is the difference between their 1000th terms?

100 Signup and view all the answers

What is the 29th term of an AP consisting of 50 terms with the 3rd term as 12 and the last term as 106?

38 Signup and view all the answers

What is the formula for the sum of the first n terms of an arithmetic progression (AP)?

$S = \frac{n(2a + (n - 1)d)}{2}$ Signup and view all the answers

If the first term of an AP is 3 and the common difference is 2, what is the 5th term of the sequence?

11 Signup and view all the answers

If there are 10 terms in an AP, what is the value of n in the formula for the sum of the first n terms?

10 Signup and view all the answers

Which statement is true regarding the sum of the first n terms of an AP?

It can be expressed as the average of the first and last terms multiplied by the number of terms. Signup and view all the answers

What is the last term of an AP if the first term is 5, common difference is 3, and n equals 4?

17 Signup and view all the answers

For which scenario is the formula $S=\frac{n(2a + (n - 1)d)}{2}$ most useful?

When the common difference is not provided. Signup and view all the answers

How can you interpret the formula $S = \frac{(a + l)n}{2}$ in the context of an AP?

As providing the total value of all terms in the AP. Signup and view all the answers

What does 'l' represent in the formula $S=\frac{(a + l)n}{2}$?

The last term of the AP Signup and view all the answers

What is the formula for the sum of the first n positive integers?

$\frac{n(n + 1)}{2}$ Signup and view all the answers

What is the value of S1000, using the derived formula for the sum?

500500 Signup and view all the answers

What is the common difference (d) for the arithmetic progression defined by $a_n = 3 + 2n$?

2 Signup and view all the answers

If the first term of an arithmetic progression is 5 and the common difference is 2, what is the 10th term?

23 Signup and view all the answers

Using the formula to compute the sum of the first n terms, S24, with a=5 and d=2, what is the calculated value?

672 Signup and view all the answers

What is the last term (l) when finding the sum of the first 1000 integers?

1000 Signup and view all the answers

In the sequence generated by the formula $a_n = 3 + 2n$, what is the first term?

5 Signup and view all the answers

To find S24 for the arithmetic series represented by $a_n = 3 + 2n$, which of these values is NOT used in the calculation?

Signup and view all the answers

How many terms of the AP: 9, 17, 25,... must be taken to give a sum of 636?

28 Signup and view all the answers

If the first term of an AP is 5, the last term is 45, and the sum is 400, what is the common difference?

6 Signup and view all the answers

Given the first term of an AP is 17, the last term is 350, and the common difference is 9, how many terms are there?

40 Signup and view all the answers

What is the sum of the first 22 terms of an AP where the common difference d = 7 and the 22nd term is 149?

1430 Signup and view all the answers

For an AP where the second term is 14 and the third term is 18, what is the sum of the first 51 terms?

600 Signup and view all the answers

If the sum of the first 7 terms of an AP is 49 and that of 17 terms is 289, what is the sum of the first n terms?

90 Signup and view all the answers

If the sum of the first n terms of an AP is defined by the equation 4n − n^2, what is the first term S1?

4 Signup and view all the answers

What is the sum of the first 15 multiples of 8?

640 Signup and view all the answers

What is the total distance the competitor has to run to collect all ten potatoes?

100 m Signup and view all the answers

In the given arithmetic progression 121, 117, 113,..., what is the common difference?

-4 Signup and view all the answers

If the sum of the third and the seventh terms of an AP is 6, what can be inferred about their average?

It is 3 Signup and view all the answers

In the ladder problem, if the bottom rung is 45 cm and the top is 25 cm, what does the uniform decrease in length imply?

The rungs decrease uniformly. Signup and view all the answers

How many rungs are there if the top and bottom rungs are 2 m apart and spaced 25 cm apart?

8 Signup and view all the answers

What is true about the value of x such that the sum of the house numbers before it equals the sum after it?

It is the median house number. Signup and view all the answers

If the third term of an arithmetic progression is 4, what could be the possible values of the first term if the common difference is 2?

3 Signup and view all the answers

In the case of the potato race, how far is the second potato from the starting point?

8 m Signup and view all the answers

Study Notes

Arithmetic Progressions (APs)

AP is a sequence where the difference between consecutive terms is constant.
Common difference (d) can be positive, negative, or zero, affecting the sequence's growth.

Finding Missing Terms in APs

To find missing elements, use the formula for the nth term: ( a_n = a + (n-1)d ).
Example sequences to find missing terms:
- ( i ) 2, ___, 26: Missing term is 14
- ( ii ) ___, 13, ___, 3: Missing terms are 7 and 1
- ( iii ) 5, ___, ___, 9: Missing terms are 6 and 7
- ( iv ) -4, ___, ___, ___, ___, 6: Missing terms are -2, 0, 2, 4
- ( v ) ___, 38, ___, ___, ___, -22: Missing terms are 16, 8, 0

Common AP Problems

Determining which term is equal to a specific value, e.g., finding which term of AP 3, 8, 13, ... equals 78.
Finding the number of terms in an AP based on the last term:
- Example: From 7, 13, 19,... to 205 - calculate using ( n = ((l - a)/d) + 1 ).

Checking Membership in APs

To check if a number like -150 is part of an AP, determine if it can be expressed as ( a_n = a + (n-1)d ) for integers ( n ).

Determining Specific Terms and Sums

If given 11th and 16th terms of an AP, find the 31st term using ( d ) and a formula for specific term calculation.
The formula for the sum of the first ( n ) terms of an AP is: ( S_n = \frac{n}{2} (2a + (n - 1)d) ) or ( S_n = \frac{n}{2} (a + l) ).

Examples of Finding AP Properties

Use the sum formulas to resolve specific cases:
- Example: Sum of 50 terms with a known 3rd term of 12 and a last term of 106 to find 29th term.
- If the 3rd and 9th terms are known, use them to find values of ( d ) and ( a ) for zeros in the series.

Advanced Patterns and Values

To solve equations involving term sums, set equalities based on known terms.
For consistent differences in two APs, evaluate their term differences using the established common difference.

Specific Requests

For any value of ( n ) where terms in differing APs equalize, use the nth term formulas to solve.
Utilize AP properties like the uniform spacing of elements to calculate total distances or sums.

Solving for Unique Counts and Conditions

The number of terms meeting specific conditions can be determined through derived formulas.
Examples include determining how many three-digit numbers are divisible by 7 through sequence formulas.

Practical Examples

Determine total distance in a given scenario with potatoes placed in a linear sequence, demonstrating real-world applications of AP.
Constructing ladders or exact spacing problems involving arithmetic sequences for practical applications.

Summary of Sum of Sequences

The sum formulation for natural numbers or specific multiplicative sequences (like multiples of 4) can derive straightforwardly from the n-th term equations.

By mastering these fundamentals of arithmetic progressions, one can approach various mathematical problems involving sequences with confidence.

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Description

Test your skills in finding missing terms in various arithmetic progressions. This quiz includes a series of sequences where you will need to fill in the blanks. Challenge yourself with these AP problems and enhance your understanding of sequences.