TThe \[{{4}^{th}}\],\[{{42}^{nd}}\] and last term of an AP are 0, -95 and -125 respectively. Find the first term and the number of terms.
Answer
381.6k+ views
Hint: Consider any AP whose first term is ‘a’ and the common difference is ‘d’. Write \[{{n}^{th}}\] term of AP as \[{{a}_{n}}=a+\left( n-1 \right)d\]. Write equations for terms of an AP and solve them using elimination method to get the value of a, n and d.
Complete step by step answer:
We have an AP whose \[{{4}^{th}},{{42}^{nd}}\] and last term is 0, -95 and -125. We have to find the first term of AP and the number of terms in AP.
Let’s assume that the first term of AP is ‘a’ and the common difference is ‘d’.
We know that we can write the \[{{n}^{th}}\] term of AP as \[{{a}_{n}}=a+\left( n-1 \right)d\].
Substituting \[n=4\] in the above equation, we have \[{{a}_{4}}=a+\left( 4-1 \right)d\]. Thus, we have \[a+3d=0.....\left( 1 \right)\].
Substituting \[n=42\] in the above equation, we have \[{{a}_{42}}=a+\left( 42-1 \right)d\]. Thus, we have \[a+41d=-95.....\left( 2 \right)\].
Subtracting equation (1) from equation (2), we have \[a+41d-\left( a+3d \right)=-95-0\].
Thus, we have \[38d=-95\Rightarrow d=-\dfrac{95}{38}=-2.5\].
Substituting the value \[d=-2.5\] in equation (1), we have \[a+3\left( -2.5 \right)=0\].
Thus, we have \[a=-7.5\].
We know that the last term of this AP is -125. Let’s assume that there are ‘x’ terms in this AP.
Thus, we have \[{{a}_{x}}=a+\left( x-1 \right)d=-125\].
Substituting \[a=-7.5,d=-2.5\] in the above formula, we have \[-7.5+\left( x-1 \right)\left( -2.5 \right)=-125\].
Simplifying the above equation, we have \[\left( x-1 \right)\left( -2.5 \right)=-117.5\].
Thus, we have \[x-1=\dfrac{-117.5}{-2.5}=47\].
So, we have \[x=47+1=48\].
Hence, the first term of this AP is -7.5 and the number of terms is 48.
Note: One must clearly know the definition of AP. Arithmetic Progression (AP) is the sequence of numbers in which the difference of two consecutive terms is a constant. We can also solve these linear equations by substitution method. We can check if the calculated solutions are correct or not by substituting the values in the equations and checking if they satisfy the equations or not.
Complete step by step answer:
We have an AP whose \[{{4}^{th}},{{42}^{nd}}\] and last term is 0, -95 and -125. We have to find the first term of AP and the number of terms in AP.
Let’s assume that the first term of AP is ‘a’ and the common difference is ‘d’.
We know that we can write the \[{{n}^{th}}\] term of AP as \[{{a}_{n}}=a+\left( n-1 \right)d\].
Substituting \[n=4\] in the above equation, we have \[{{a}_{4}}=a+\left( 4-1 \right)d\]. Thus, we have \[a+3d=0.....\left( 1 \right)\].
Substituting \[n=42\] in the above equation, we have \[{{a}_{42}}=a+\left( 42-1 \right)d\]. Thus, we have \[a+41d=-95.....\left( 2 \right)\].
Subtracting equation (1) from equation (2), we have \[a+41d-\left( a+3d \right)=-95-0\].
Thus, we have \[38d=-95\Rightarrow d=-\dfrac{95}{38}=-2.5\].
Substituting the value \[d=-2.5\] in equation (1), we have \[a+3\left( -2.5 \right)=0\].
Thus, we have \[a=-7.5\].
We know that the last term of this AP is -125. Let’s assume that there are ‘x’ terms in this AP.
Thus, we have \[{{a}_{x}}=a+\left( x-1 \right)d=-125\].
Substituting \[a=-7.5,d=-2.5\] in the above formula, we have \[-7.5+\left( x-1 \right)\left( -2.5 \right)=-125\].
Simplifying the above equation, we have \[\left( x-1 \right)\left( -2.5 \right)=-117.5\].
Thus, we have \[x-1=\dfrac{-117.5}{-2.5}=47\].
So, we have \[x=47+1=48\].
Hence, the first term of this AP is -7.5 and the number of terms is 48.
Note: One must clearly know the definition of AP. Arithmetic Progression (AP) is the sequence of numbers in which the difference of two consecutive terms is a constant. We can also solve these linear equations by substitution method. We can check if the calculated solutions are correct or not by substituting the values in the equations and checking if they satisfy the equations or not.
Recently Updated Pages
Define absolute refractive index of a medium

Find out what do the algal bloom and redtides sign class 10 biology CBSE

Prove that the function fleft x right xn is continuous class 12 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Find the values of other five trigonometric ratios class 10 maths CBSE

Find the values of other five trigonometric functions class 10 maths CBSE

Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Which one of the following places is unlikely to be class 8 physics CBSE

Select the word that is correctly spelled a Twelveth class 10 english CBSE

Difference Between Plant Cell and Animal Cell

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

What is the past tense of read class 10 english CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Elucidate the structure of fructose class 12 chemistry CBSE

What is pollution? How many types of pollution? Define it
