Question

Q4. The 3rd term of an arithmetic series, A. is 19 The sum of the first 10 terms of A is 290 Find the 10th term of A.​

Answer 1

Answer:

The 10ᵗʰ term of the arithmetic series A is 47.

Step-by-step-explanation:

We have given that,

For an arithmetic series A,

• t₃ = 19
• S₁₀ = 290

We have to find the 10ᵗʰ term of A.

Now, we know that,

tₙ = a + ( n - 1 ) * d - - - [ Formula ]

⇒ t₃ = a + ( 3 - 1 ) * d

⇒ 19 = a + 2d

⇒ a + 2d = 19

⇒ a = 19 - 2d

⇒ a = - 2d + 19 - - - ( 1 )

Now, we know that,

Sₙ = ( n / 2 ) [ 2a + ( n - 1 ) * d ] - - - [ Formula ]

⇒ S₁₀ = ( 10 / 2 ) [ 2a + ( 10 - 1 ) * d ]

⇒ 290 = 5 [ 2 * ( - 2d + 19 ) + 9d ] - - - [ From ( 1 ) ]

⇒ 290 ÷ 5 = 2 ( - 2d + 19 ) + 9d

⇒ - 4d + 38 + 9d = 58

⇒ - 4d + 9d = 58 - 38

⇒ 5d = 20

⇒ d = 20 ÷ 5

⇒ d = 4

By substituting d = 4 in equation ( 1 ), we get,

a = - 2d + 19 - - - ( 1 )

⇒ a = - 2 * 4 + 19

⇒ a = - 8 + 19

⇒ a = 11

Now, we have to find t₁₀,

tₙ = a + ( n - 1 ) * d

⇒ t₁₀ = 11 + ( 10 - 1 ) * 4

⇒ t₁₀ = 11 + 9 * 4

⇒ t₁₀ = 11 + 36

⇒ t₁₀ = 47

∴ The 10ᵗʰ term of the arithmetic series A is 47.

Answer 2

ɪɴꜰᴏʀᴍᴀᴛɪᴏɴ ᴘʀᴏᴠɪᴅᴇᴅ ᴡɪᴛʜ ᴜꜱ :

• The 3rd term of an arithmetic series A is 19
• The sum of the first 10 terms of A is 290

ᴡʜᴀᴛ ᴡᴇ ʜᴀᴠᴇ ᴛᴏ ᴄᴀʟᴄᴜʟᴀᴛᴇ :

• 10th term of A?

Using Formulas :

★ General term of an A.P. or Series is calculated by,

• tₙ = a + (n - 1)d

Here,

• a is first term
• n is number of terms
• d is common difference

★ Sum of n terms is calculated by,

• S = n / 2 [2a + (n - 1) d]

Here,

• S is sum of the terms
• n is number of terms
• d is common difference
• a is first term

ᴘᴇʀꜰᴏʀᴍɪɴɢ ᴄᴀʟᴄᴜʟᴀᴛɪᴏɴꜱ :

We have :

• S = 290
• 3rd term = 19

Putting the values in general term of an A.P. :

➺ t₃ = a + (3 - 1) d [Remember that n would be 3]

➺ t₃ = a + ( 2 ) d

Now, here substituting the value of t₃ in it,

➺ 19 = a + ( 2 ) d

➺ 19 = a + 2 × d

➺ 19 = a + 2d

➺ a = 19 - 2d

Putting the values in formula of sum of an A.P. or series :

Remember that here we would be taking number of terms (n) as 10.

➺ 290 = 10/2 [ 2a + (10 - 1) d ]

➺ 290 = 5 [ 2a + (10 - 1) d ]

Now, here substituting the value of a,

➺ 290 = 5 [ 2a + (10 - 1) d ]

➺ 290 = 5 [ 2 (19 - 2d) + (10 - 1) d ]

➺ 290 = 5 [ 2 × (19 - 2d) + (10 - 1) d ]

➺ 290 = 5 [ 38 - 4d + (10 - 1) d ]

➺ 290 = 5 [ 38 - 4d + 10d - d ]

➺ 290 = 5 [ 38 + 6d - d ]

➺ 290 = 5 [ 38 + 5d ]

➺ 290 = 5 × [ 38 + 5d ]

➺ 290 = 190 + 25d

➺ 25d = 290 - 190

➺ 25d = 100

➺ d = 100 / 25

➺ d = 20 / 5

➺ d = 4

Finding out first term of the A.P. or series :

➺ 19 = a + ( 2 ) 4

➺ 19 = a + 2 × 4

➺ 19 = a + 8

➺ a = 19 - 8

➺ a = 11

According to the question,

Here we are calculating the 10th term by substituting the values of first term (a) , common difference (d) in the formula of calculating the nth term.

➺ t₁₀ = 11 + (10 - 1) 4

➺ t₁₀ = 11 + (9) 4

➺ t₁₀ = 11 + 9 × 4

➺ t₁₀ = 11 + 36

➺ t₁₀ = 47

• Henceforth, 10th term of A is 47 !

ᴇxᴛʀᴀ ɪɴꜰᴏʀᴍᴀᴛɪᴏɴ :

• Arithmetic progression (A.P.) is a sequence (series) in which each term can be found by adding a certain quantity to its preceding term
• Difference between two consecutive terms is called common difference
• Progression means it's a type of sequence in which each term is related to its predecessor and successor.

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