How many terms are identical in the following two A.P. 2,4,6, up to 100 terms and 3,6,9, up to 80 terms.?
First AP: 2, 4 , 6... up-to 100 terms.So a= 2d= 2n= 100So tn= a+(n-1)dby solving we get,First series 2,4, 6 .....200.Second AP, 3, 6, 9.. up-to 80 terms.So a= 3d= 3n= 80So tn= a+(n-1)dby solving we get,Second Series 3, 6, 9, .....240.After analysing the two series we have common terms between between the two series, starting from 6 to 200 with a common difference of 6. 200/6= not an integer.So new AP starts from 6 to 198Now,tn=a+(n-1)dtn=198a=6d=6198=6+(n-1)6192=(n-1)6n-1=32n=33Hence total number of identical terms are 33.
Four numbers in AP.whose sum is 20 and sum of their squares is 180.find the numbers.?
Four numbers are in A.P. therefore the terms are : a-3d,a-d,a+d,a+3d .By 1st conditionSum of the squares of number is =180The sum of the term is =20(a-3d)+(a-d)+(a+d)+(a+3d) =204a =20a=5By 2nd condition(a-3d)^2+(a-d)^2+(a+d)^2+(a+3d)^2 =180Substitute the value of a and solve the equation100+20d^2=18020d^2=180–100d^2=80/20d^2=4d=2 or d=-2a=5 d=2 or -2When d=2a-3d= -1 and a-d= 3a+3d= 11 and a+d= 7When d=-2a-3d= 11 and a-d= 7a+3d= -1 and a+d= 3The numbers are -1 ,3,7,11 and -1,3,7,11.
Physics Question: Heat flow by conduction?
Q-dot = energy/sec = - [k A/ (- L)] (dT) watt Q-dot = dT / (L/k A) watt --------------------------------------... direction of length and dT fall is opposite in conduction ================== compare with current - resistance equivalent, dT temperature difference is just like potential didderence which makes Q-dot to flow. (L/k A) is thermal resistance offered by material I = V / R ------------------------ R = (L/k A) >>> proportional to L, (1/A) and (1/k) thermal resistivity -------------------------------- now rods are in parallel, same dV (potential) 1/R = 1/R1+1/R2 >>> overall resistance 1/R = k1 A/L + k2 A / L >>> same L same A (pi r^2) 1/R = A(k1+k2) / L Rc = L / A (k1+k2) ----------------------------------- composite heat flow Qc-dot = dT / Rc -------------------------------- in parallel, you calculate total resistance then divide voltage with R(total) for find I composite (I1+I2) current ---------------------------------- Qc-dot [L /A (k1+k2)] = dT L = A (k1+k2)* dT /[Qc-dot] ---------------------------------- therm conduc of steel is shown different on diff sites, what is in your book??? L = pi*(0.0125)^2 * [217+66.9] (105) /22.5 L = 0.65 meter Rods are 65 cm long each if answer is different then k of steel to be checked
Find the sum of all 3 digit numbers divisible by 7?
Smallest 3 digit number is 100 and largest 3 digit number is 999. Smallest 3 digit number divisible by 7 is 105. Largest 3 digit number divisible by 7 is 994 So we have an arithmetic sequence as follows - (105, 112, 119...994) a = 105 and d = 7 The sequence can be rewritten as [7(15), 7(16), 7(17)...7(142)] Adding all of this results in 7(15 + 16 + 17 + ...142) Recall Euler summation identity that sum of successive terms is n(n + 1) /2 So we have 7 [ 142(143)/2 - 14(15/2)] = 7[ 71(143) - 7(15)] = 70336
The sum of d three numbers in an AP is 21 and their product is 315.find d numbers?
In AP, First number = x Second number = x + n Third number = x + 2*n Given, [1] x + (x + n) + (x + 2*n) = 21 [2] x * (x + n) * (x + 2*n) = 315 Solve for the two unknowns (x and n) in two independent equations gives two possibilities: (x = 5 and n = 2) or (x = 9 or n = -2) This means the 3 numbers in sequence are: (5, 7, 9) or (9, 7, 5). Since you only want the 3 numbers (regardless of sequence), the numbers are 5, 7, and 9.
Sum of four numbers in arithmetic progression is 24 and their product is 945.find the numbers.?
Captain Matticus' answer is very original and, hence, very commendable. As an alternative, I offer the following : Let the 4 numbers be : a - 3d, a - d, a + d, a + 3d. Their sum is : 4a = 24 ∴ a = 6 ∴ the above numbers now become : 6 - 3d, 6 - d, 6 + d, 6 + 3d ...... (1) Now use the second condition to find d. That will complete the answer. __________________________________