Question

two aps have same common difference if difference their 50 term is 50 find the difference of their 100 terms is​

Answer 1

Answer:

$\bigstar{\bold{Difference\:between\:the\:100th\:terms=50}}$

Step-by-step explanation:

$\Large{\underline{\underline{\bf{Given:}}}}$

• The common difference of two A.P is the same
• Difference of their 50th terms is 50

$\Large{\underline{\underline{\bf{To\:Find:}}}}$

• Difference of their 100th terms

$\Large{\underline{\underline{\bf{Solution:}}}}$

→ Let the common difference of the two A.Ps be d

→ Let the first term of first A.P be a

→ Let the first term of second A.P be A

→ The 50th term of the first A.P is given by

  a₅₀ = a₁ + (50 - 1) × d

  a₅₀ = a₁ + 49 d ---(1)

→ The 50th term of the second A.P is given by

   A₅₀ = A₁ + (50 - 1) × d

   A₅₀ = A₁ + 49 d -----(2)

→ By given,

  equation 2 - equation 1 = 50

→ Hence,

  A₁ + 49 d - (a₁ + 49 d )= 50

→ Cancelling 49 d we get,

  A₁ - a₁ = 50 -----(1)

→ Now the difference between their hundredth terms is given by

  A₁ + 99d - a₁ - 99d

→ Cancelling 99 d we get

  Difference between 100th terms = A₁ - a₁

→ Substitute the value from equation 1

  Difference between 100th terms = 50

→ Hence the difference between the 100th terms of the A.P is 50

$\boxed{\bold{Difference\:between\:the\:100th\:terms=50}}$

$\Large{\underline{\underline{\bf{Notes:}}}}$

→ The nth term of an A.P is given by

   $\sf{a_n=a_1+(n-1)\times d}$

→ The common difference of an A.P is given by

  $\sf{d=\dfrac{a_m-a_n}{m-n} }$

Answer 2

Solution :

$\sf{Let \: the \: common \: difference \: of \: two \: A.Ps \: be \: d} \\ \\ \sf{Then, \: their \: 50^{th} \: terms \: be} \\ \\ \sf{ a_{50} = a_1 + 49d } \: \\ \\ \sf{ b_{50}= b_1+ 49d} \\ \\ \sf{Given \: that}, \\ \\ \implies\sf{a_{50} -b_{50} = 50} \\ \\ \implies\sf{i.e., (a_1+ 49d) - (b_1+ 49d) = 50} \\ \\ \implies \sf{a_1 - b_1= 50} \\ \\ \sf{Now, \: difference \: between \: their \: 100^{th} \: terms} \\ \\ \implies\sf{a_{100} - b_{100} }\\ \\ \implies\sf{(a_1+99d) - (b_1+ 99d)} \\ \\ \implies\sf{a_1 - b_1= 50}$