Predicting What Comes Next: Sequences and Progressions | EOT

Question 8

The sum of the 4th4^{\text{th}} and 8th8^{\text{th}} terms of an AP is 24 and the sum of the 6th6^{\text{th}} and 10th10^{\text{th}} terms is 44. Find the first three terms of the AP.

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Solution

We will use the formula for the nthn^{\text{th}} term of an AP, which is an=a+(n−1)da_n = a + (n-1)d.

Step 1 — Form the first equation

Let the first term of the AP be aa. Let the common difference be dd. The 4th4^{\text{th}} term is a+(4−1)da + (4-1)d. So, the 4th4^{\text{th}} term is a+3da + \mathbf{3d}. The 8th8^{\text{th}} term is a+(8−1)da + (8-1)d. So, the 8th8^{\text{th}} term is a+7da + \mathbf{7d}. The sum of the 4th4^{\text{th}} and 8th8^{\text{th}} terms is 24\mathbf{24}. Let's write this as an equation.

(a+3d)+(a+7d)=24(a + 3d) + (a + 7d) = 24

2a+10d=242a + 10d = 24

a+5d=12a + 5d = 12

a+5d=12(Equation 1)\boxed{a + 5d = 12 \quad \text{(Equation 1)}}

Step 2 — Form the second equation

The 6th6^{\text{th}} term is a+(6−1)da + (6-1)d. So, the 6th6^{\text{th}} term is a+5da + \mathbf{5d}. The 10th10^{\text{th}} term is a+(10−1)da + (10-1)d. So, the 10th10^{\text{th}} term is a+9da + \mathbf{9d}. The sum of the 6th6^{\text{th}} and 10th10^{\text{th}} terms is 44\mathbf{44}. Let's write this as another equation.

(a+5d)+(a+9d)=44(a + 5d) + (a + 9d) = 44

2a+14d=442a + 14d = 44

a+7d=22a + 7d = 22

a+7d=22(Equation 2)\boxed{a + 7d = 22 \quad \text{(Equation 2)}}

Step 3 — Solve for 'a' and 'd'

Now we have two simple equations. Let's subtract Equation 1 from Equation 2.

(a+7d)−(a+5d)=22−12(a + 7d) - (a + 5d) = 22 - 12

a+7d−a−5d=10a + 7d - a - 5d = 10

2d=102d = 10

d=102d = \frac{10}{2}

d=5d = 5 Now, substitute the value of dd into Equation 1.

a+5(5)=12a + 5(5) = 12

a+25=12a + 25 = 12

a=12−25a = 12 - 25

a=−13a = -13

a=−13,d=5\boxed{a = -13, d = 5}

Step 4 — Find the first three terms

We know the first term aa and the common difference dd. The first term is aa. The second term is a+da + d. The third term is a+2da + 2d.

First term: −13\mathbf{-13}

Second term: −13+5=−8-13 + 5 = \mathbf{-8}

Third term: −13+2(5)=−13+10=−3-13 + 2(5) = -13 + 10 = \mathbf{-3}

Answer

The first three terms of the AP are -13, -8, -3.

More questions in EOT

Q1

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Q2

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Q3

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Q4

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Q5

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Q6

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Q7

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Q8

The sum of the 4th4^{\text{th}} and 8th8^{\text{th}} terms of an AP is 24 and the sum of the 6th6^{\text{th}} and 10th10^{\text{th}} terms is 44. Find the first three terms of the AP.

Q9

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Q10

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Q11

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Q12

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Q13

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Q14

Suppose P1=1P_1 = 1, P2=2P_2 = 2 and for n>2n > 2, Pn=P1+P2+⋯+Pn−1+1P_n = P_1 + P_2 + \dots + P_{n-1} + 1. Find the values of P1,P2,…,P8P_1, P_2, \dots, P_8. Can you find a simpler recursive formula for PnP_n? Can you give an explicit formula?

Q15

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