if you want to remove an article from website contact us from top.

# find the sum of the first 34 terms of the arithmetic progression whose first term is 6 and whose common difference is 8/3.

Category :

### Mohammed

Guys, does anyone know the answer?

get find the sum of the first 34 terms of the arithmetic progression whose first term is 6 and whose common difference is 8/3. from screen.

## The sum of the first n terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another AP whose first term is 40 and the common difference is 6 . Find n .

Click here👆to get an answer to your question ✍️ The sum of the first n terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another AP whose first term is 40 and the common difference is 6 . Find n .

Question

## The sum of the first n terms of an AP whose first term is 8 and the common difference is 20 is equal to the sum of first 2n terms of another AP whose first term is 40 and the common difference is 6. Find n.

Medium Open in App

Updated on : 2022-09-05

Solution Verified by Toppr

Correct option is A)

Let S

n ​

be the sum of n terms of an AP with first term a=8 and common difference d=20. Then,

S n ​ = 2 n ​ (2a+(n−1)d)⇒S n ​ = 2 n ​ (2×8+(n−1)×20)⇒S n ​ = 2 n ​ (16+20n−20) ⇒S n ​ = 2 n ​

(20n−4)              ...1

similarly Let S 2n ​

be the sum of 2n terms of an AP with first term a1=30 and common difference d1=8. Then,

S 2n ​ = 2 2n ​ (2a1+(2n−1)d1)⇒S 2n ​ = 2 2n ​ (2×30+(n−1)×8)⇒S 2n ​ = 2 2n ​ (60+8n−8) ⇒S 2n ​

=n(8n+52)           ...2

According to question S

n ​ =S 2n ​ 2 n ​ (20n−4)=n(8n+52) ⇒10n−2=8n+52 ⇒2n=54 ⇒n=27

64 26

स्रोत : www.toppr.com

## Arithmetic Progression (AP)

An arithmetic progression is a sequence where the differences between every two consecutive terms are the same. In an arithmetic progression, each number is obtained by adding a fixed number to the previous term.

## Arithmetic Progression

An arithmetic progression (AP) is a sequence where the differences between every two consecutive terms are the same. In this type of progression, there is a possibility to derive a formula for the nth term of the AP. For example, the sequence 2, 6, 10, 14, … is an arithmetic progression (AP) because it follows a pattern where each number is obtained by adding 4 to the previous term. In this sequence, nth term = 4n-2. The terms of the sequence can be obtained by substituting n=1,2,3,... in the nth term. i.e.,

When n = 1, first term = 4n-2 = 4(1)-2 = 4-2=2

When n = 2, second term = 4n-2 = 4(2)-2 = 8-2=6

When n = 3, thirs term = 4n-2 = 4(3)-2 = 12-2=10

In this article, we will explore the concept of arithmetic progression, the formula to find its nth term, common difference, and the sum of n terms of an AP. We will solve various examples based on arithmetic progression formula for a better understanding of the concept.

## What is Arithmetic Progression?

We can define an arithmetic progression (AP) in two ways:

An arithmetic progression is a sequence where the differences between every two consecutive terms are the same.

An arithmetic progression is a sequence where each term, except the first term, is obtained by adding a fixed number to its previous term.

For example, 1, 5, 9, 13, 17, 21, 25, 29, 33, ... has

a = 1 (the first term)

d = 4 (the "common difference" between terms)

In general an arithmetic sequence can be written like: {a, a+d, a+2d, a+3d, ... }.

Using the above example we get: {a, a+d, a+2d, a+3d, ... } = {1, 1+4, 1+2×4, 1+3×4, ... } = {1, 5, 9, 13, ... }

### Arithmetic Progression Definition

Arithmetic progression is defined as the sequence of numbers in algebra such that the difference between every consecutive term is the same. It can be obtained by adding a fixed number to each previous term.

## Arithmetic Progression Formula

For the first term 'a' of an AP and common difference 'd', given below is a list of arithmetic progression formulas that are commonly used to solve various problems related to AP:

Common difference of an AP: d = a2 - a1 = a3 - a2 = a4 - a3 = ... = an - an-1

nth term of an AP: an = a + (n - 1)d

Sum of n terms of an AP: Sn = n/2(2a+(n-1)d) = n/2(a + l), where l is the last term of the arithmetic progression.

### AP Formula

The image below shows the formulas related to arithmetic progression:

## Common Terms Used in Arithmetic Progression

From now on, we will abbreviate arithmetic progression as AP. Here are some more AP examples:

6, 13, 20, 27, 34, . . . .

91, 81, 71, 61, 51, . . . .

π, 2π, 3π, 4π, 5π,…

-√3, −2√3, −3√3, −4√3, −5√3,…

An AP generally is shown as follows: a1, a2, a3, . . . It involves the following terminology.

### First Term of Arithmetic Progression:

As the name suggests, the first term of an AP is the first number of the progression. It is usually represented by a1 (or) a. For example, in the sequence 6,13,20,27,34, . . . . the first term is 6. i.e., a1=6 (or) a=6.

### Common Difference of Arithmetic Progression:

We know that an AP is a sequence where each term, except the first term, is obtained by adding a fixed number to its previous term. Here, the “fixed number” is called the “common difference” and is denoted by 'd' i.e., if the first term is a1, then: the second term is a1+d, the third term is a1+d+d = a1+2d, and the fourth term is a1+2d+d= a1+3d and so on. For example, in the sequence 6,13,20,27,34,. . . , each term, except the first term, is obtained by addition of 7 to its previous term. Thus, the common difference is, d=7. In general, the common difference is the difference between every two successive terms of an AP. Thus, the formula for calculating the common difference of an AP is: d = an - an-1

## Nth Term of Arithmetic Progression

The general term (or) nth term of an AP whose first term is 'a' and the common difference is 'd' is found by the formula an=a+(n-1)d. For example, to find the general term (or) nth term of the sequence 6,13,20,27,34,. . . ., we substitute the first term, a1=6, and the common difference, d=7 in the formula for the nth term formula. Then we get, an =a+(n-1)d = 6+(n-1)7 = 6+7n-7 = 7n -1. Thus, the general term (or) nth term of this sequence is: an = 7n-1. But what is the use of finding the general term of an AP? Let us see.

### AP Formula for General Term

We know that to find a term, we can add 'd' to its previous term. For example, if we have to find the 6th term of 6,13,20,27,34, . . ., we can just add d=7 to the 5th term which is 34. 6th term = 5th term + 7 = 34+7 = 41. But what if we have to find the 102nd term? Isn’t it difficult to calculate it manually? In this case, we can just substitute n=102 (and also a=6 and d=7 in the formula of the nth term of an AP). Then we get:

an = a+(n-1)d a102 = 6+(102-1)7 a102 = 6+(101)7 a102 = 713

Therefore, the 102nd term of the given sequence 6,13,20,27,34,.... is 713. Thus, the general term (or) nth term of an AP is referred to as the arithmetic sequence explicit formula and can be used to find any term of the AP without finding its previous term.

स्रोत : www.cuemath.com

## What is the number of terms in the progression when the first term of an arithmetic sequence is 8 and the last term is 34, with the sum of the first six terms being 58?

Answer (1 of 6): Answer for the number of terms is 40. To solve for the common difference, d, consider the first 6 terms where the sum of 58 with the first term as 8 and the last term is the 6th term. Determine the last term, an in the formula for sum, S=n/2(a1+an) considering the last term for...

What is the number of terms in the progression when the first term of an arithmetic sequence is 8 and the last term is 34, with the sum of the first six terms being 58?

Ad by Amazon Web Services (AWS)

AWS is how.

AWS removes the complexity of building, training, and deploying machine learning models at any scale.

Sort Joshua Johansson 3y

Let d be the common difference and n be the number of terms

Sum of the first six terms:

6/2(2*8+5*d)=58 3(16+5d)=58 48+15d=58 15d=10 d=2/3 Solve for n: 34=8+(n-1)d 26=(n-1)2/3 39=n-1 Therefore, n=40 Related questions

What is the sum of the first 50 terms of an arithmetic progression, given that the 15th term is 34 and the sumof the first 8 terms is 20?

The sum of the first 10 terms in an arithmetic progression is 50 and the sum of its first eight terms is 56. How can I find the first term and the common ratio?

What is the sum of the first 5 terms of an arithmetic progression if the first term is 7 and the last term is 55?

The last term of an arithmetic sequence is 207, the common difference is 3, and the number of terms is 14. What is the first term?

In an arithmetic progression, its 5th term is 3, and the sum of the first 10 terms is 105/4. Which term is 0?

Sum of the first six terms = 58

i.e.

6/2* [ 2(8) + (6–1)d ] = 58

3[ 16+5d] = 58 48 +15d = 58 d = 2/3

and the last term = 34

i.e. 8 + (n-1)*2/3 = 34 (n-1)*2/3 = 26 n-1 = 39 n =40 Sponsored by Aspose

What is Aspose.OCR for C++ library?

OCR API capable of extracting text from BMP, JPEG and other images having different fonts and styles.

Murali Krishna

I was a Mathematics TEACHER teaching MATHEMATICS for stuAuthor has 6.5K answers and 4.2M answer views1y

AP a=8 an=34 S6=58 Sn =n/2 (2a+(n-1)d) S6=3[2×8+5d] =58 16+5d =58/3

5d=58/3 --16/1 =58-48=10 /3

5d=10/3 d=10/15 =2/3 a=8 an=8+(n-1) 2/3 =34 (n-1)=26×3/2 =39 n=40 Number of terms =40 Verification a=8 d=2/3 S6 =3(16+5×2/3) =58 John R. Clymer

PhD in Electrical Engineering & Mathematics, Arizona State University (Graduated 1971)Author has 611 answers and 306K answer viewsNov 5

A6 = 8 + d(5) = l

sum = 6/2(8 + (8+5d)) = 58

3(16+5d) = 58 48 + 15d = 58 15d = 10 d=1/2

An = 8 + 1(n-1)/2 = 34

n/2 = 34 - 7.5 = 63/2

n = 63

check: A63 = 16/2 + 62/2 = 78/2 = 34

Related questions

If the third term of an arithmetic sequence is 12 and the eight term is 27, what is the fifth term?

The sum of the first eight terms of an arithmetic progression is 56 and the sum of its first 20 terms is 260. What is the first term and the common difference of the A.P.?

The 4th term of an arithmetic sequence is 8 and the 6th term is 28. What is the first term?

If the first arithmetic sequence is -9 and the common difference is 8, what is the sum of the first 12 terms?

In an arithmetic sequence, the eleventh term is T11= 17 and the fiftieth term is T50 = 95. If A is the first term of the sequence and D its common difference, how do we find A and D?

Calvin L.

S 6 = 6 2 [2(8)+5d]=58 S6=62[2(8)+5d]=58 5d= 58 3 −16= 10 3 5d=583−16=103 d= 2 3 d=23 34=8+ 2 3 (n−1) 34=8+23(n−1) n−1=39⟹n=40 n−1=39⟹n=40 terms.

Do you feel like the internet manipulates people?

Claim your own free plot of land in the metaverse and setup rules to protect your mental health.

Catalino Lansangan

Former Civil and Structural Inspector at Stantec Consulting (2010–2016)Author has 206 answers and 281.8K answer views3y

Answer for the number of terms is 40.

To solve for the common difference, d, consider the first 6 terms where the sum of 58 with the first term as 8 and the last term is the 6th term.

Determine the last term, an in the formula for sum, S=n/2(a1+an) considering the last term for the first 6 terms with the sum of 58 is the 6th term:

58=6/2(8+an) 58=3(8+an)=24+3an 58-24=3an 34=3an

an=34/3=11–1/3 or 11.333 is the 6th term or a6

Solve for d, as the common difference:

Let an=a6, where the common difference is always the same no matter how many terms involved.

a6=a1+(n-1)d 11.333=8+(6–1)d 11.333–8=5d 3.333=5d. d= Shashwat Sourav

BS in Engineering Science, Indian Institute of Science Education and Research, Bhopal (IISER-B) (Expected 2024)Upvoted by

Lalit Narayan Vyas

, MSc. , Mathematics, BU Bhopal (2001)Author has 866 answers and 612.5K answer views1y

Related

The 6th term of an AP is 6 and the 16th term is 14. What is the 27th term?

Hope this helps you Ruchi.