If the sum of the first 6 terms of an AP is 3 | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(A) Let $a$ be the first term and $d$ be the common difference of the $AP$.The sum of $n$ terms of an $AP$ is given by $S_n = \frac{n}{2}[2a + (n-1)d]

Vedclass · Accepted Answer

$100$

Vedclass · Answer

(A) Let $a$ be the first term and $d$ be the common difference of the $AP$.The sum of $n$ terms of an $AP$ is given by $S_n = \frac{n}{2}[2a + (n-1)d]$.Given $S_6 = 36$:$\frac{6}{2}[2a + (6-1)d] = 36$$3[2a + 5d] = 36$$2a + 5d = 12$ ....$(i)$Given $S_{16} = 256$:$\frac{16}{2}[2a + (16-1)d] = 256$$8[2a + 15d] = 256$$2a + 15d = 32$ ....$(ii)$Subtracting equation $(i)$ from equation $(ii)$:$(2a + 15d) - (2a + 5d) = 32 - 12$$10d = 20$$d = 2$Substituting $d = 2$ into equation $(i)$:$2a + 5(2) = 12$$2a + 10 = 12$$2a = 2$$a = 1$Now,find the sum of the first $10$ terms $(S_{10})$:$S_{10} = \frac{10}{2}[2a + (10-1)d]$$S_{10} = 5[2(1) + 9(2)]$$S_{10} = 5[2 + 18]$$S_{10} = 5 	imes 20 = 100$.Thus,the sum of the first $10$ terms is $100$.

If the sum of the first $6$ terms of an $AP$ is $36$ and that of the first $16$ terms is $256,$ find the sum of the first $10$ terms.

Similar Questions

Determine whether the following sequence is an $A.P.$ or not. (Assume that the pattern continues.) If it is an $A.P.$,find its $n^{th}$ term: $5, 11, 17, 23, \ldots$

For any $A.P.$,$T_{30} - T_{20} = \ldots \ldots \ldots \ldots$

Find the $12^{th}$ term of the $A.P.$ $9, 13, 17, 21, \ldots$

Verify that each of the following is an $AP$,and then write its next three terms.
$0, \frac{1}{4}, \frac{1}{2}, \frac{3}{4}, \ldots$

For an $A.P.$,the $10^{th}$ term is $52$ and the $16^{th}$ term is $82$. Find the $n^{th}$ term and the $32^{nd}$ term of the $A.P.$

Vedclass Test Series

Exam Paper Generator

Online Exam Module