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Arithmetic & Geometric Sequences

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Quick recall

  • AP: constant difference. GP: constant ratio.
  • The nth term and the sum of n terms are different questions.

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Formula Cards

1. AP — nth term

an=a1+(n−1)da_n=a_1+(n-1)d

d is the constant difference.

2. AP — sum

Sn=n2[2a1+(n−1)d]S_n=\frac{n}{2}[2a_1+(n-1)d]

3. GP — nth term

an=a1rn−1a_n=a_1r^{n-1}

r is the constant ratio.

4. GP — finite sum

Sn=a1(rn−1)r−1S_n=\frac{a_1(r^n-1)}{r-1}

For r ≠ 1. If r = 1, the sum is n times the first term.

5. GP — infinite sum

S∞=a11−rS_\infty=\frac{a_1}{1-r}

Only when the absolute value of r is less than 1.

Worked Examples

Example 1

Question: An AP starts at 3 with common difference 4. Find its fifth term.

Step 1: Count the increments.

n−1=5−1=4n-1=5-1=4

Step 2: Add four increments to the first term.

a5=3+4(4)=19a_5=3+4(4)=19

Answer: 19.

Example 2

Question: A GP starts at 2 with ratio 3. Find its fourth term.

Step 1: Use exponent n−1.

a4=2×34−1a_4=2\times3^{4-1}

Step 2: Evaluate.

a4=2×27=54a_4=2\times27=54

Answer: 54.

Example 3

Question: Find the sixth term of the AP 2, 5, 8, 11, …

Step 1: Identify the common difference.

d=5−2=3d=5-2=3

Step 2: Apply the nth-term formula.

a6=2+(6−1)(3)=17a_6=2+(6-1)(3)=17

Answer: 17.

Example 4

Question: Find the sum of the first five terms of the GP 1, 2, 4, 8, …

Step 1: Identify a₁, r, n.

a1=1,r=2,n=5a_1=1,\qquad r=2,\qquad n=5

Step 2: Use the finite-sum formula.

S5=1(25−1)2−1=31S_5=\frac{1(2^5-1)}{2-1}=31

Answer: 31.

Example 5

Question: Find the sum of the first 20 terms of the AP 3, 7, 11, …

Step 1: Identify the first term and difference.

a1=3,d=4a_1=3,\qquad d=4

Step 2: Substitute into the sum formula.

S20=202[2(3)+19(4)]S_{20}=\frac{20}{2}[2(3)+19(4)]

Step 3: Evaluate.

S20=10(82)=820S_{20}=10(82)=820

Answer: 820. The previous answer of 1,060 was an arithmetic error.

Drill

Try each question before opening its answer.

Question 1

Find the 10th term of AP: 5, 8, 11...

  • Show answer

    32

Question 2

Find the sum of first 10 terms of AP: 2, 4, 6...

  • Show answer

    110

Question 3

Find the 5th term of GP: 3, 6, 12...

  • Show answer

    48

Question 4

Find sum to infinity of GP: 8, 4, 2, 1...

  • Show answer

    16

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