Arithmetic Mean & Arithmetic Series
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The document defines arithmetic mean and arithmetic series. Arithmetic mean is calculated by adding all terms of a sequence and dividing by the total number of terms. It provides examples of inserting arithmetic means between given terms. An arithmetic series is the sum of an arithmetic sequence. The sum is calculated by multiplying the number of terms by the average of the first and last terms. It provides examples of calculating the sum of arithmetic series.
![ARITHMETIC SERIES
Sn =
𝑛
2
(a1 + an)
Formulae
Sn =
𝑛
2
[2a1 + (n - 1) d]](https://image.slidesharecdn.com/mathproject-150816091404-lva1-app6891/85/Arithmetic-Mean-Arithmetic-Series-13-320.jpg)
Arithmetic Mean & Arithmetic Series
- 3. ARITHMETIC MEAN Finding a certain number of terms between two given terms of an arithmetic sequence The terms between any two nonconsecutive terms of an arithmetic sequence are known as arithmetic means
- 4. ARITHMETIC MEAN Arithmetic Mean = 𝑥1 + 𝑥2 + 𝑥3 + … + 𝑥𝑛 𝑛 Formula
- 5. ARITHMETIC MEAN Insert 4 arithmetic means between 5 and 25. • Since we are required to insert 4 terms, then there will be 6 terms in all. • Let a1=5 and a6=25, we will insert a2, a3, a4, a5 as shown below. 5, a2, a3, a4, a5, 25 Example 1
- 6. ARITHMETIC MEAN •Use a6=a1+5d to solve for d. 25= 5+5d 5d=25-5 5d=20 d=4 Example 1
- 7. ARITHMETIC MEAN Using the value of d, we can now get the values of a2, a3, a4, a5, thus: a2=9, a3=13, a4=17, a5=21 Example 1 5, 9, 13, 17, 21, 25
- 8. ARITHMETIC MEAN Insert 4 arithmetic means between -24 and -4. • Since we are required to insert 4 terms, then there will be 6 terms in all. • Let a1=-24 and a6=-4, we will insert a2, a3, a4, a5 as shown below. -24, a2, a3, a4, a5, -4 Example 2
- 9. ARITHMETIC MEAN Use a6=a1+5d to solve for d. -4= -24+5d 5d=4-24 5d=-20 d=-4 Example 2
- 10. ARITHMETIC MEAN Using the value of d, we can now get the values of a2, a3, a4, a5, thus: a2=-20, a3=-16, a4=-12, a5=-8 Example 2 -24, -20, -16, -12, -8, -4
- 12. ARITHMETIC SERIES It is the sum of an arithmetic sequence The sum of an arithmetic series is found by multiplying the number of terms times the average of the first and last terms.
- 13. ARITHMETIC SERIES Sn = 𝑛 2 (a1 + an) Formulae Sn = 𝑛 2 [2a1 + (n - 1) d]
- 14. ARITHMETIC SERIES 3 + 8 + 13 + … + 73 • a1 = 3 • an = 73 • d = 5 • n = ? • Sn = ? Example 1
- 15. ARITHMETIC SERIES Sn = 𝑛 2 (a1 + an) S15 = 15 2 (3 + 73) S15 = 15 2 (76) S15 = 15 (38) S15 = 490 Example 1
- 16. ARITHMETIC SERIES What is the sum of the first 10 terms in the series: 3, 7, 11, … • a1 = 3 •d = 4 • n = 10 • an = ? • Sn = ? Example 2
- 17. ARITHMETIC SERIES an = a1 + (n – 1) d a10 = 3 + (10 – 1) 4 a10 = 3 + (9) 4 a10 = 3 + 36 a10 = 39 Example 2
- 18. ARITHMETIC SERIES Sn = 𝑛 2 (a1 + an) S10 = 10 2 (3 + 39) S10 = 5 (42) S10 = 210 Example 2