Insert 6

Insert four harmonic means between 3/7 and 3/19

Correct answer:

a₂ =  0.355
a₃ =  0.303
a₄ =  0.2643
a₅ =  0.2344

Step-by-step explanation:

$$\begin{gathered} a_1=3\mathrm{/}7={\displaystyle \frac{3}{7}}\doteq 0.4286 \\ {a}_6=4\mathrm{/}19={\displaystyle \frac{4}{19}}\doteq 0.2105 \\ {b}_1=1\mathrm{/}{a}_1=1\mathrm{/}0.4286={\displaystyle \frac{7}{3}}=2\frac{1}{3}\doteq 2.3333 \\ {b}_6=1\mathrm{/}{a}_6=1\mathrm{/}0.2105={\displaystyle \frac{19}{4}}=4\frac{3}{4}=4.75 \\ d=(b_6-b_1)\mathrm{/}(6-1)=(4.75-2.3333)\mathrm{/}(6-1)={\displaystyle \frac{29}{60}}\doteq 0.4833 \\ {b}_2=b_1+d=2.3333+0.4833={\displaystyle \frac{169}{60}}=2\frac{49}{60}\doteq 2.8167 \\ {a}_2=1\mathrm{/}{b}_2=1\mathrm{/}2.8167={\displaystyle \frac{60}{169}}=0.355 \end{gathered}$$

$$\begin{gathered} b_3=b_2+d=2.8167+0.4833={\displaystyle \frac{33}{10}}=3\frac{3}{10}=3.3 \\ {a}_3=1\mathrm{/}{b}_3=1\mathrm{/}3.3={\displaystyle \frac{10}{33}}=0.303 \end{gathered}$$

$$\begin{gathered} b_4=b_3+d=3.3+0.4833={\displaystyle \frac{227}{60}}=3\frac{47}{60}\doteq 3.7833 \\ {a}_4=1\mathrm{/}{b}_4=1\mathrm{/}3.7833={\displaystyle \frac{60}{227}}=0.2643 \end{gathered}$$

$$\begin{gathered} b_5=b_4+d=3.7833+0.4833={\displaystyle \frac{64}{15}}=4\frac{4}{15}\doteq 4.2667 \\ {a}_5=1\mathrm{/}{b}_5=1\mathrm{/}4.2667={\displaystyle \frac{15}{64}}=0.2344 \end{gathered}$$

Try another example

Related math problems and questions:

• Insert 7
  Insert five harmonic means between 3 and 18
• Insert 5
  Insert five harmonic means between 1/2 and 1/26
• HP - harmonic progression
  Determine the 10th term of the harmonic progression 6,4,3,…
• Harmonic series
  Insert four members between 5/3 and 5/11 to form harmonic series (means).
• Harmonic mean
  If x, y, z form a harmonic progression, then y is the harmonic mean of x and z. Find the harmonic mean of the numbers 6 and 5.
• Harmonic mean
  Harmonic means of 6 and 12
• Insert 4
  Insert three arithmetic means between 3 and 63.
• Insert 3
  Insert five arithmetic progression members between -7 and 3/2.
• Harmonic 4
  The harmonic mean of -6 and 5.
• Insert AP member
  Insert arithmetic means between 75 and 180.
• HP - harmonic progression
  Determine the 8th term of the harmonic progression 2, 4/3, 1,…
• Sort fractions
  Which is larger 3/7, 3/8, 3/9, 3/6 =
• Mailbox
  Mailbox is opening at regular intervals 3 times a day. The first time is opened at 8:00 and the last at 19:00. Calculate hours when mailbox is opened during day.
• Sequence 11
  What is the nth term of this sequence 1, 1/2, 1/3, 1/4, 1/5?
• What are
  What are the four terms between 4 and 19?
• Tractors
  Fields go plow two tractors with various performances. The first tractor plows the whole field in 11 hours, the second tractor plows the whole field 19 hours longer. How long take plow the whole field with two tractors?
• Car
  The car goes from point A to point B at speed 86 km/h and back 53 km/h. If it goes there and back at speed 67 km/h trip would take 10 minutes shorter. What is the distance between points A and B?