Advertisements
Advertisements
Question
If the 5th and 12th terms of an A.P. are 30 and 65 respectively, what is the sum of first 20 terms?
Advertisements
Solution
\[\text { We have }: \]
\[ a_5 = 30\]
\[ \Rightarrow a + \left( 5 - 1 \right)d = 30\]
\[ \Rightarrow a + 4d = 30 . . . (i)\]
\[\text { Also }, a_{12} = 65\]
\[ \Rightarrow a + \left( 12 - 1 \right)d = 65\]
\[ \Rightarrow a + 11d = 65 . . . . . (ii)\]
\[\text { Solving (i) and (ii), we get }: \]
\[7d = 35\]
\[ \Rightarrow d = 5\]
\[\text { Putting the value of d in (i), we get }: \]
\[a + 4 \times 5 = 30\]
\[ \Rightarrow a = 10\]
\[ \therefore S_{20} = \frac{20}{2}\left[ 2 \times 10 + (20 - 1) \times 5 \right]\]
\[ \Rightarrow S_{20} = 10\left[ 2 \times 10 + (20 - 1) \times 5 \right]\]
\[ \Rightarrow S_{20} = 1150\]
APPEARS IN
RELATED QUESTIONS
How many terms of the A.P. -6 , `-11/2` , -5... are needed to give the sum –25?
Find the sum to n terms of the A.P., whose kth term is 5k + 1.
Sum of the first p, q and r terms of an A.P. are a, b and c, respectively.
Prove that `a/p (q - r) + b/q (r- p) + c/r (p - q) = 0`
Insert five numbers between 8 and 26 such that the resulting sequence is an A.P.
if `(a^n + b^n)/(a^(n-1) + b^(n-1))` is the A.M. between a and b, then find the value of n.
Let < an > be a sequence. Write the first five term in the following:
a1 = 1, an = an − 1 + 2, n ≥ 2
The nth term of a sequence is given by an = 2n2 + n + 1. Show that it is not an A.P.
Which term of the A.P. 84, 80, 76, ... is 0?
Which term of the A.P. 4, 9, 14, ... is 254?
How many numbers of two digit are divisible by 3?
If < an > is an A.P. such that \[\frac{a_4}{a_7} = \frac{2}{3}, \text { find }\frac{a_6}{a_8}\].
The sum of three terms of an A.P. is 21 and the product of the first and the third terms exceeds the second term by 6, find three terms.
The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Find the sum of the following serie:
2 + 5 + 8 + ... + 182
Find the sum of all integers between 50 and 500 which are divisible by 7.
Find the sum of all even integers between 101 and 999.
Find the sum of all those integers between 100 and 800 each of which on division by 16 leaves the remainder 7.
Solve:
1 + 4 + 7 + 10 + ... + x = 590.
Find the r th term of an A.P., the sum of whose first n terms is 3n2 + 2n.
The sum of first 7 terms of an A.P. is 10 and that of next 7 terms is 17. Find the progression.
If \[\frac{b + c}{a}, \frac{c + a}{b}, \frac{a + b}{c}\] are in A.P., prove that:
bc, ca, ab are in A.P.
If x, y, z are in A.P. and A1 is the A.M. of x and y and A2 is the A.M. of y and z, then prove that the A.M. of A1 and A2 is y.
A man arranges to pay off a debt of Rs 3600 by 40 annual instalments which form an arithmetic series. When 30 of the instalments are paid, he dies leaving one-third of the debt unpaid, find the value of the first instalment.
A piece of equipment cost a certain factory Rs 600,000. If it depreciates in value, 15% the first, 13.5% the next year, 12% the third year, and so on. What will be its value at the end of 10 years, all percentages applying to the original cost?
A carpenter was hired to build 192 window frames. The first day he made five frames and each day thereafter he made two more frames than he made the day before. How many days did it take him to finish the job?
We know that the sum of the interior angles of a triangle is 180°. Show that the sums of the interior angles of polygons with 3, 4, 5, 6, ... sides form an arithmetic progression. Find the sum of the interior angles for a 21 sided polygon.
Write the common difference of an A.P. whose nth term is xn + y.
If the sums of n terms of two arithmetic progressions are in the ratio 2n + 5 : 3n + 4, then write the ratio of their m th terms.
Write the value of n for which n th terms of the A.P.s 3, 10, 17, ... and 63, 65, 67, .... are equal.
The first and last terms of an A.P. are 1 and 11. If the sum of its terms is 36, then the number of terms will be
In the arithmetic progression whose common difference is non-zero, the sum of first 3 n terms is equal to the sum of next n terms. Then the ratio of the sum of the first 2 n terms to the next 2 nterms is
Write the quadratic equation the arithmetic and geometric means of whose roots are Aand G respectively.
If there are (2n + 1) terms in an A.P., then prove that the ratio of the sum of odd terms and the sum of even terms is (n + 1) : n
The product of three numbers in A.P. is 224, and the largest number is 7 times the smallest. Find the numbers
A man accepts a position with an initial salary of Rs 5200 per month. It is understood that he will receive an automatic increase of Rs 320 in the very next month and each month thereafter. Find his salary for the tenth month
Let Sn denote the sum of the first n terms of an A.P. If S2n = 3Sn then S3n: Sn is equal to ______.
If the sum of n terms of a sequence is quadratic expression then it always represents an A.P
