Arithmetic Problem Solving

This discussion of mathematics subjects will explore the problem of arithmetic sequences and series.

The most frequently asked or questioned questions are finding the nth th term of a sequence, determining the difference between the tribes, determining the number of tribes and finding the first tribe

But before we go into practice questions, we will first understand what is the sequence and the arithmetic series. Because in addition to the sequence and the arithmetic series, we also know the sequence and the geometry series. But in this discussion, our focus is on the sequence and the arithmetic series.

Arithmetic Rows and Arrays

Once described in the next explanation, we are expected to understand what the sequence or series and the differences are and also to know the purpose of the sequence and the arithmetic series. Besides, we can understand which is called tribe and different value.

What is a Row?

A sequence is an array of numbers formed according to a particular sequence. Each number in a row is a tribe in a row.
Example:

• 1, 2, 3, 4, 5,6,7  (Number 1 is the first term, the number 2 is the second term and so on)
• 2, 5, 8, 11, 14,17 (The number 8 is the third tribe, the number 17 is the sixth tribe).
• 14, 12, 10, 8, 6, 4, 2 (the number 12 is the second tribe while the number 10 is the third tribe dst).

So obviously, that sequence is a set of numbers that have a particular pattern, while the numbers that form a sequence with a specified pattern is named after the tribe. There are acting as the first, second, third etc.

What is a Series?

the sum of the tribes of a sequence is called a series. If U₁,U₂,U₃,…..U_n
then : U₁ + U₂ + U₃ +… +U_n is the sequence.

Example:

1 + 2 + 3 + 4 +… + .U_n

2 + 4 + 6 + 8 +… + .U_n

What is an Arithmetic sequence?

The arithmetic sequence is a sequence that has the value of the difference between two successive tribes always fixed. The difference of two consecutive terms is called a different value, symbolized byb.

In the arithmetic sequence, the order of the differences between one term and the next is constant. In other words, we just add the same value every time.

Example:

1, 4, 7, 10, 13, 16, 19, 22, 25, ...

The sequence has 3 different values ​​between one term and the next.

In general, we can write the arithmetic sequence:

{a, a+b, a+2b, a+3b, ... }

Where:

• a is the first tribe,
• b is a different value.

Arithmetic Row Formulas

1. To find the nth Tribe:

U_n = a + (n - 1)b
where:

• U_n: nth term
• a: first tribe
• b: different value
• n: many tribes

2. To find different values:

b = U_n-U_(n-1)
where:

• b is a different value
• U_n: nth term

3. To find the Middle Tribe
We can look for the middle tribe that has n odd tribe (number of odd tribe) where is known the first and last tribe , then used the formula:

U_t = a + U_n 2
where :

• U_tis the middle term
• a is the first term
• U_n is the nth term (in this case acting as the last tribe)

But if to find the middle tribe whose condition is only known to the first tribe, the number of n tribes and different values, then the formula:

U_t = a + (n-1)b 2where :

• U_t is the middle term
• a is the first term
• n states the number of tribes
• b represents a different value

What is Arithmetic Arrangement

Arithmetic sequence is the sum of the usual arithmetic sequences marked with the plus sign (+).

Example :

• 2 + 4 + 6 + 8 + 10
• 3 + 6 + 9 + 12 + 15

To find the sum of an arithmetic series, the formula is used:

S_n = n 2 (a+U_n)
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S_n = n 2 (2a + (n-1)b) where :

• S_n denotes the sum of the nth term
• a is the first term/li>
• U_n denotes the value of the nth term
• b represents a different value
• n states the number of tribes

Exercises

Problem No.1

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An arithmetic sequence has an odd number of tribes. If the first tribe 6 and the last tribe is 22, then the middle tribe is:
a. 10
b. 14
c. 16
d. 18

Discussion

a = 6
U_n = 22
U_t= a + U_n 2 = 22 + 62 = 14

Answer : b


Problem No.2

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There is an arithmetic sequence of eight tribes. If the first term and the difference value are 4. What is the middle term?
a. 5
b. 7
c. 9
d. 11

Discussion

a = 2
b = 2
n = 7
U_t= a + (n-1)b 2U_t= a + (n-1)b 2 = 4 + (8-1)2 2 = 11

Answer : d


Problem No.3

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Given an arithmetic sequence: 3, 8, 13, 18,  23, .........U_n. Find the nth term formula in the arithmetic sequence:
a. U_n = 5n -1
a. U_n = 5n -2
c. U_n = 5n + 1
d. U_n = 5n + 3

Discussion :

a = 3
b = 5
U_n= a + (n-1)b
U_n= 3 + (n-1)5 = 3 + 5n - 5 = 5n-2

Answer : b


Problem No.4

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Given U₂ + U₄ = 12 dan U₃ + U₅ = 16, then the 7th term of the sequence is
a. 15
b. 14
c. 12
d. 10

Discussion

From the sum of the 2nd and 4th tribes :
(1) U₂ +U₄ = 12
⇒ (a + b) + (a + 3b) = 12
⇒ 2 a + 4b = 12
⇒ a + 2b = 6

From summation the 3rd and 5th tribes:
(2) U₃ + U₅ = 16
⇒ (a + 2b) + (a + 4b) = 16
⇒ 2a + 6b = 16
⇒ a + 3b = 8

The next step, we will substituting the equation 1 to equation 2 :
a + 2b = 6
a = 6 – 2b.... substitution to equation (2)

Equation (2):
a + 3b = 8
⇒ 6 – 2b + 3b = 8
⇒ 6 + b = 8
⇒ b = 2

Because b = 2, then a = 6 – 2(2) = 6 – 4 = 2.

Thus, the first term of the sequence is 2 and the 7th term of the arithmetic sequence is:
U₇ = a + 6b
⇒ U₇ = 2 + 6(2) ⇒ U₇ = 14

Answer: b


Problem No.5

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In an arithmetic sequence the second tribe is 5 and the fifth is 14. So what is the sum of the first 10 terms of the arithmetic sequence?
a. 210
b. 300
c. 430
d. 155

Discussion:

Second Tribe :
⇒ U₂ = 5
⇒ a + b = 5
⇒ a = 5 - b...(Equation 1)

The Fifth Tribe :
⇒ U₅ = 14
⇒ a + 4b = 14...(Equation 2)

Substitution of Equation 1 to Equation 2
⇒ a + 4b = 14
⇒ 5 - b + 4b = 14
⇒ 3b = 9
⇒ b = 3
So a = 5 -b
⇒ a = 5 - 3 = 2

The first 10 terms:
⇒ S_n= n 2 (a+U_n)
⇒ S₁₀= 10 2 (a+U₁₀)
⇒ S₁₀= 5 (a + a + 9b)
⇒ S₁₀= 5 (2 + 2 + 9.3)
⇒ S₁₀= 155

Answer: d