# 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.

Last updated date: 26th Mar 2023

•

Total views: 306k

•

Views today: 6.83k

Answer

Verified

306k+ 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

If a spring has a period T and is cut into the n equal class 11 physics CBSE

A planet moves around the sun in nearly circular orbit class 11 physics CBSE

In any triangle AB2 BC4 CA3 and D is the midpoint of class 11 maths JEE_Main

In a Delta ABC 2asin dfracAB+C2 is equal to IIT Screening class 11 maths JEE_Main

If in aDelta ABCangle A 45circ angle C 60circ then class 11 maths JEE_Main

If in a triangle rmABC side a sqrt 3 + 1rmcm and angle class 11 maths JEE_Main

Trending doubts

Difference Between Plant Cell and Animal Cell

Write an application to the principal requesting five class 10 english CBSE

Ray optics is valid when characteristic dimensions class 12 physics CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Write the 6 fundamental rights of India and explain in detail

Write a letter to the principal requesting him to grant class 10 english CBSE

List out three methods of soil conservation

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

Epipetalous and syngenesious stamens occur in aSolanaceae class 11 biology CBSE